Changeset 7eb3aa2 in sasview for src/sans/models
- Timestamp:
- Feb 17, 2014 11:59:48 AM (11 years ago)
- Branches:
- 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
- Children:
- c6ca23d
- Parents:
- 71aa9ac
- Location:
- src/sans/models/media
- Files:
-
- 3 added
- 1 edited
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img/pringle-vs-cylinder.png (added)
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img/pringle_eqn_1.jpg (added)
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img/pringle_eqn_2.jpg (added)
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model_functions.html (modified) (15 diffs)
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src/sans/models/media/model_functions.html
r083e993 r7eb3aa2 7 7 <ul style="margin-top: 0in;" type="disc"> 8 8 <li><a href="#Introduction"><b>Introduction</b></a></li> 9 <li><a href="#Shapes"><b>Shapes</b></a>: <a href="#SphereModel">SphereModel (Magnetic 2D Model)</a>, <a href="#BinaryHSModel">BinaryHSModel</a>, <a href="#FuzzySphereModel">FuzzySphereModel</a>, <a href="#RaspBerryModel">RaspBerryModel</a>, <a href="#CoreShellModel">CoreShellModel (Magnetic 2D Model)</a>, <a href="#Core2ndMomentModel">Core2ndMomentModel</a>, <a href="#CoreMultiShellModel">CoreMultiShellModel (Magnetic 2D Model)</a>, <a href="#VesicleModel">VesicleModel</a>, <a href="#MultiShellModel">MultiShellModel</a>, <a href="#OnionExpShellModel">OnionExpShellModel</a>, <a href="#SphericalSLDModel">SphericalSLDModel</a>, <a href="#LinearPearlsModel">LinearPearlsModel</a>, <a href="#PearlNecklaceModel">PearlNecklaceModel</a> , <a href="#CylinderModel">CylinderModel (Magnetic 2D Model)</a>, <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, <a href="#CoreShellBicelleModel">CoreShellBicelleModel</a>,<a href="#HollowCylinderModel">HollowCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, <a href="#FlexibleCylinderModel">FlexCylEllipXModel</a>, <a href="#StackedDisksModel">StackedDisksModel</a>, <a href="#ParallelepipedModel">ParallelepipedModel (Magnetic 2D Model)</a>, <a href="#CSParallelepipedModel">CSParallelepipedModel</a>, <a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>, <a href="#BarBellModel">BarBellModel</a>, <a href="#CappedCylinderModel">CappedCylinderModel</a>, <a href="#EllipsoidModel">EllipsoidModel</a>, <a href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, <a href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a>, <a href="#LamellarModel">LamellarModel</a>, <a href="#LamellarFFHGModel">LamellarFFHGModel</a>, <a href="#LamellarPSModel">LamellarPSModel</a>, <a href="#LamellarPSHGModel">LamellarPSHGModel</a>, <a href="#LamellarPCrystalModel">LamellarPCrystalModel</a>, <a href="#SCCrystalModel">SCCrystalModel</a>, <a href="#FCCrystalModel">FCCrystalModel</a>, <a href="#BCCrystalModel">BCCrystalModel</a>.</li> 10 <li><a href="#Shape-Independent"><b>Shape-Independent</b></a>: <a href="#Absolute%20Power_Law">AbsolutePower_Law</a>, <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, <a href="#BroadPeakModel">BroadPeakModel,<span><span style="text-decoration: underline;"><span style="color: blue;">CorrLength</span></span></span><span>,</span></a> <a href="#DABModel">DABModel</a>, <a href="#Debye">Debye</a>, <a href="#Number_Density_Fractal">FractalModel</a>, <a href="#FractalCoreShell">FractalCoreShell</a>, <a href="#GaussLorentzGel">GaussLorentzGel</a>, <a href="#Guinier">Guinier</a>, <a href="#GuinierPorod">GuinierPorod</a>, <a href="#Lorentz">Lorentz</a>, <a href="#Mass_Fractal">MassFractalModel</a>, <a href="#MassSurface_Fractal">MassSurfaceFractal</a>, <a href="#Peak%20Gauss%20Model">PeakGaussModel</a>, <a href="#Peak%20Lorentz%20Model">PeakLorentzModel</a>, <a href="#Poly_GaussCoil">Poly_GaussCoil</a>, <a href="#PolymerExclVolume">PolyExclVolume</a>, <a href="#PorodModel">PorodModel</a>, <a href="#RPA10Model">RPA10Model</a>, <a href="#StarPolymer">StarPolymer</a>, <a href="#Surface_Fractal">SurfaceFractalModel</a>, <a href="#TeubnerStreyModel">Teubner Strey</a>, <a href="#TwoLorentzian">TwoLorentzian</a>, <a href="#TwoPowerLaw">TwoPowerLaw</a>, <a href="#UnifiedPowerRg">UnifiedPowerRg</a>, <a href="#LineModel">LineModel</a>, <a href="#ReflectivityModel">ReflectivityModel</a>, <a href="#ReflectivityIIModel">ReflectivityIIModel</a>, <a href="#GelFitModel">GelFitModel</a>.</li> 11 <li><a href="#Model"><b>Customized Models</b></a>: <a href="#testmodel">testmodel</a>, <a href="#testmodel_2">testmodel_2</a>, <a href="#sum_p1_p2">sum_p1_p2</a>, <a href="#sum_Ap1_1_Ap2">sum_Ap1_1_Ap2</a>, <a href="#polynomial5">polynomial5</a>, <a href="#sph_bessel_jn">sph_bessel_jn</a>.</li> 12 <li><a href="#Structure_Factors"><b>Structure Factors</b></a>: <a href="#HardsphereStructure">HardSphereStructure</a>, <a href="#SquareWellStructure">SquareWellStructure</a>, <a href="#HayterMSAStructure">HayterMSAStructure</a>, <a href="#StickyHSStructure">StickyHSStructure</a>.</li> 9 <li><a href="#Shapes"><b>Shapes</b></a>: 10 <ul> 11 <li>Sphere based:<br/> 12 <a href="#SphereModel">SphereModel (Magnetic 2D Model)</a>, 13 <a href="#BinaryHSModel">BinaryHSModel</a>, 14 <a href="#FuzzySphereModel">FuzzySphereModel</a>, 15 <a href="#RaspBerryModel">RaspBerryModel</a>, 16 <a href="#CoreShellModel">CoreShellModel (Magnetic 2D Model)</a>, 17 <a href="#Core2ndMomentModel">Core2ndMomentModel</a>, 18 <a href="#CoreMultiShellModel">CoreMultiShellModel (Magnetic 2D Model)</a>, 19 <a href="#VesicleModel">VesicleModel</a>, 20 <a href="#MultiShellModel">MultiShellModel</a>, 21 <a href="#OnionExpShellModel">OnionExpShellModel</a>, 22 <a href="#SphericalSLDModel">SphericalSLDModel</a>, 23 <a href="#LinearPearlsModel">LinearPearlsModel</a>, 24 <a href="#PearlNecklaceModel">PearlNecklaceModel</a> 25 </li> 26 <li>Cylinder based:<br/> 27 <a href="#CylinderModel">CylinderModel (Magnetic 2D Model)</a>, 28 <a href="#CoreShellCylinderModel">CoreShellCylinderModel</a>, 29 <a href="#CoreShellBicelleModel">CoreShellBicelleModel</a>, 30 <a href="#HollowCylinderModel">HollowCylinderModel</a>, 31 <a href="#FlexibleCylinderModel">FlexibleCylinderModel</a>, 32 <a href="#FlexibleCylinderModel">FlexCylEllipXModel</a>, 33 <a href="#StackedDisksModel">StackedDisksModel</a>, 34 <a href="#EllipticalCylinderModel">EllipticalCylinderModel</a>, 35 <a href="#BarBellModel">BarBellModel</a>, 36 <a href="#CappedCylinderModel">CappedCylinderModel</a>, 37 <a href="#PringleModel">PringleModel</a> 38 </li> 39 <li>Parallelpipeds:<br/> 40 <a href="#ParallelepipedModel">ParallelepipedModel (Magnetic 2D Model)</a>, 41 <a href="#CSParallelepipedModel">CSParallelepipedModel</a> 42 </li> 43 <li>Ellipsoids:<br/> 44 <a href="#EllipsoidModel">EllipsoidModel</a>, 45 <a href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, 46 <a href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a> 47 </li> 48 <li>Lamellar:<br/> 49 <a href="#LamellarModel">LamellarModel</a>, 50 <a href="#LamellarFFHGModel">LamellarFFHGModel</a>, 51 <a href="#LamellarPSModel">LamellarPSModel</a>, 52 <a href="#LamellarPSHGModel">LamellarPSHGModel</a> 53 </li> 54 <li>Paracrystals:<br/> 55 <a href="#LamellarPCrystalModel">LamellarPCrystalModel</a>, 56 <a href="#SCCrystalModel">SCCrystalModel</a>, 57 <a href="#FCCrystalModel">FCCrystalModel</a>, 58 <a href="#BCCrystalModel">BCCrystalModel</a>.</li> 59 </li> 60 </ul> 61 <li><a href="#Shape-Independent"><b>Shape-Independent</b></a>: 62 <a href="#Absolute%20Power_Law">AbsolutePower_Law</a>, 63 <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, 64 <a href="#BroadPeakModel">BroadPeakModel</a>, 65 <a href="#CorrLength">CorrLength</a>, 66 <a href="#DABModel">DABModel</a>, 67 <a href="#Debye">Debye</a>, 68 <a href="#Number_Density_Fractal">FractalModel</a>, 69 <a href="#FractalCoreShell">FractalCoreShell</a>, 70 <a href="#GaussLorentzGel">GaussLorentzGel</a>, 71 <a href="#Guinier">Guinier</a>, 72 <a href="#GuinierPorod">GuinierPorod</a>, 73 <a href="#Lorentz">Lorentz</a>, 74 <a href="#Mass_Fractal">MassFractalModel</a>, 75 <a href="#MassSurface_Fractal">MassSurfaceFractal</a>, 76 <a href="#Peak%20Gauss%20Model">PeakGaussModel</a>, 77 <a href="#Peak%20Lorentz%20Model">PeakLorentzModel</a>, 78 <a href="#Poly_GaussCoil">Poly_GaussCoil</a>, 79 <a href="#PolymerExclVolume">PolyExclVolume</a>, 80 <a href="#PorodModel">PorodModel</a>, 81 <a href="#RPA10Model">RPA10Model</a>, 82 <a href="#StarPolymer">StarPolymer</a>, 83 <a href="#Surface_Fractal">SurfaceFractalModel</a>, 84 <a href="#TeubnerStreyModel">Teubner Strey</a>, 85 <a href="#TwoLorentzian">TwoLorentzian</a>, 86 <a href="#TwoPowerLaw">TwoPowerLaw</a>, 87 <a href="#UnifiedPowerRg">UnifiedPowerRg</a>, 88 <a href="#LineModel">LineModel</a>, 89 <a href="#ReflectivityModel">ReflectivityModel</a>, 90 <a href="#ReflectivityIIModel">ReflectivityIIModel</a>, 91 <a href="#GelFitModel">GelFitModel</a>.</li> 92 93 <li><a href="#Model"><b>Customized Models</b></a>: 94 <a href="#testmodel">testmodel</a>, 95 <a href="#testmodel_2">testmodel_2</a>, 96 <a href="#sum_p1_p2">sum_p1_p2</a>, 97 <a href="#sum_Ap1_1_Ap2">sum_Ap1_1_Ap2</a>, 98 <a href="#polynomial5">polynomial5</a>, 99 <a href="#sph_bessel_jn">sph_bessel_jn</a>.</li> 100 101 <li><a href="#Structure_Factors"><b>Structure Factors</b></a>: 102 <a href="#HardsphereStructure">HardSphereStructure</a>, 103 <a href="#SquareWellStructure">SquareWellStructure</a>, 104 <a href="#HayterMSAStructure">HayterMSAStructure</a>, 105 <a href="#StickyHSStructure">StickyHSStructure</a>.</li> 106 13 107 <li><a href="#References"><b>References</b></a></li> 14 108 </ul> … … 2662 2756 <p>Kratky, O. and Porod, G., J. Colloid Science, 4, 35, 1949.</p> 2663 2757 <p>Higgins, J.S. and Benoit, H.C., "Polymers and Neutron Scattering", Clarendon, Oxford, 1994.</p> 2664 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.21.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="ParallelepipedModel"></a><b><span style="font-size: 14pt;">ParallelepipedModel (Magnetic 2D Model) </span></b></p>2665 <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>2666 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help.2667 <p><span style="font-size: 14pt;"> </span></p>2668 <p style="text-align: center;" align="center"><img src="img/image087.jpg" alt="" width="326" height="247" /></p>2669 <p>The side of the solid must be satisfied the condition of A<B</p>2670 <p>By this definition, assuming</p>2671 <p>a = A/B<1; b=B/B=1; c=C/B>1, the form factor,</p>2672 <p style="text-align: center;" align="center"><span style="position: relative; top: 48pt;"><img src="img/image088.PNG" alt="" /></span></p>2673 <p>The contrast is defined as</p>2674 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/image089.PNG" alt="" /></span></p>2675 <p>The scattering intensity per unit volume is returned in the unit of [cm-1]; I(q) = <span style="font-family: Symbol;">f</span>P(q).</p>2676 <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>2677 <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>2678 <p style="text-align: center;" align="center"><img src="img/image090.jpg"/></p>2679 <p style="text-align: center;" align="center"><b>Figure. Definition of angles for 2D</b>.</p>2680 <p style="text-align: center;" align="center"><img src="img/image091.jpg" alt="" width="379" height="256" /></p>2681 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p>2682 <div align="center">2683 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0">2684 <tbody>2685 <tr style="height: 18.8pt;">2686 <td style="border: 1pt solid width: 107pt; height: 18.8pt;" valign="top" width="143">2687 <p>Parameter name</p>2688 </td>2689 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2690 <p>Units</p>2691 </td>2692 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2693 <p>Default value</p>2694 </td>2695 </tr>2696 <tr style="height: 18.8pt;">2697 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2698 <p>background</p>2699 </td>2700 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2701 <p>cm-1</p>2702 </td>2703 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2704 <p>0.0</p>2705 </td>2706 </tr>2707 <tr style="height: 18.8pt;">2708 <td style="border-width: medium 1pt 1pt; vertical-align: top; width: 107pt; height: 18.8pt;">2709 <p>contrast</p>2710 </td>2711 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2712 <p>Å -2</p>2713 </td>2714 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2715 <p>5e-006</p>2716 </td>2717 </tr>2718 <tr style="height: 18.8pt;">2719 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2720 <p>long_c</p>2721 </td>2722 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2723 <p>Å</p>2724 </td>2725 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2726 <p>400</p>2727 </td>2728 </tr>2729 <tr style="height: 18.8pt;">2730 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2731 <p>short_a</p>2732 </td>2733 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2734 <p>Å -2</p>2735 </td>2736 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2737 <p>35</p>2738 </td>2739 </tr>2740 <tr style="height: 18.8pt;">2741 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2742 <p>short_b</p>2743 </td>2744 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2745 <p>Å</p>2746 </td>2747 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2748 <p>75</p>2749 </td>2750 </tr>2751 <tr style="height: 18.8pt;">2752 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2753 <p>scale</p>2754 </td>2755 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> </td>2756 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2757 <p>1</p>2758 </td>2759 </tr>2760 </tbody>2761 </table>2762 </div>2763 <p style="text-align: center;" align="center"><img id="Picture 492" src="img/image092.jpg" alt="" width="455" height="351" /></p>2764 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p>2765 <p style="margin-left: 1.35in; text-indent: -0.25in;"><span style="font-family: Symbol;">·</span><span style="font-size: 7pt;"> </span><b>Validation of the parallelepiped 2D model</b></p>2766 <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>2767 <p style="text-align: center;" align="center"><img id="Picture 104" src="img/image093.gif" alt="" width="481" height="299" /></p>2768 <p style="text-align: center;" align="center"><b>Figure. Comparison between 1D and averaged 2D.</b></p>2769 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p>2770 <p>REFERENCE</p>2771 <p>Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 185-211.</p>2772 <p>Equations (1), (13-14). (in German)</p>2773 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.22.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"><a name="CSParallelepipedModel"></a>CSParallelepipedModel</span></b></p>2774 <p>Calculates the form factor for a rectangular solid with a core-shell structure. The thickness and the scattering length density of the shell or "rim" can be different on all three (pairs) of faces. The form factor is normalized by the particle volume such that P(q) = scale*<f^2>/Vol + bkg, where < > is an average over all possible orientations of the rectangular solid. An instrument resolution smeared version is also provided.</p>2775 <p>The function calculated is the form factor of the rectangular solid below. The core of the solid is defined by the dimensions ABC such that A < B < C. </p>2776 <p style="text-align: center;" align="center"><img id="Picture 38" src="img/image087.jpg" alt="" width="326" height="247" /> </p>2777 <p>There are rectangular "slabs" of thickness tA that add to the A dimension (on the BC faces). There are similar slabs on the AC (=tB) and AB (=tC) faces. The projection in the AB plane is then:</p>2778 <p style="text-align: center;" align="center"><img src="img/image094.jpg" alt="" width="334" height="277" /></p>2779 <p>The volume of the solid is:</p>2780 <p><span style="position: relative; top: 5pt;"><img src="img/image095.PNG" alt="" /></span></p>2781 <p>meaning that there are "gaps" at the corners of the solid.</p>2782 <p>The intensity calculated follows the parallelepiped model, with the core-shell intensity being calculated as the square of the sum of the amplitudes of the core and shell, in the same manner as a core-shell sphere.</p>2783 <p>For the calculation of the form factor to be valid, the sides of the solid MUST be chosen such that A < B < C. If this inequality in not satisfied, the model will not report an error, and the calculation will not be correct.</p>2784 <p>FITTING NOTES:</p>2785 <p>If the scale is set equal to the particle volume fraction, f, the returned value is the scattered intensity per unit volume, I(q) = f*P(q). However, no interparticle interference effects are included in this calculation.</p>2786 <p>There are many parameters in this model. Hold as many fixed as possible with known values, or you will certainly end up at a solution that is unphysical.</p>2787 <p>Constraints must be applied during fitting to ensure that the inequality A < B < C is not violated. The calculation will not report an error, but the results will not be correct.</p>2788 <p>The returned value is in units of [cm-1], on absolute scale.</p>2789 <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>2790 <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>2791 <p style="text-align: center;" align="center"><img id="Picture 102" src="img/image090.jpg" /></p>2792 <p style="text-align: center;" align="center"><b>Figure. Definition of angles for 2D</b>.</p>2793 <p style="text-align: center;" align="center"><img id="Picture 103" src="img/image091.jpg" alt="" width="379" height="256" /></p>2794 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented cspp against the detector plane.</p>2795 <p> TEST DATASET</p>2796 <p>This example dataset is produced by running the Macro Plot_CSParallelepiped(), using 100 data points, qmin = 0.001 Å-1, qmax = 0.7 Å-1 and the below default values. </p>2797 <div align="center">2798 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0">2799 <tbody>2800 <tr style="height: 18.8pt;">2801 <td style="border: 1pt solid width: 107pt; height: 18.8pt;" valign="top" width="143">2802 <p>Parameter name</p>2803 </td>2804 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2805 <p>Units</p>2806 </td>2807 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2808 <p>Default value</p>2809 </td>2810 </tr>2811 <tr style="height: 18.8pt;">2812 <td style="border-width: medium 1pt 1pt; vertical-align: top; width: 107pt; height: 18.8pt;">2813 <p>background</p>2814 </td>2815 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2816 <p>cm-1</p>2817 </td>2818 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2819 <p>0.06</p>2820 </td>2821 </tr>2822 <tr style="height: 18.8pt;">2823 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2824 <p>sld_pcore</p>2825 </td>2826 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2827 <p>Å -2</p>2828 </td>2829 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2830 <p>1e-006</p>2831 </td>2832 </tr>2833 <tr style="height: 18.8pt;">2834 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2835 <p>sld_rimA</p>2836 </td>2837 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2838 <p>Å -2</p>2839 </td>2840 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2841 <p>2e-006</p>2842 </td>2843 </tr>2844 <tr style="height: 18.8pt;">2845 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2846 <p>sld_rimB</p>2847 </td>2848 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2849 <p>Å -2</p>2850 </td>2851 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2852 <p>4e-006</p>2853 </td>2854 </tr>2855 <tr style="height: 18.8pt;">2856 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2857 <p>sld_rimC</p>2858 </td>2859 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2860 <p>Å -2</p>2861 </td>2862 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2863 <p>2e-006</p>2864 </td>2865 </tr>2866 <tr style="height: 18.8pt;">2867 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2868 <p>sld_solv</p>2869 </td>2870 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2871 <p>Å -2</p>2872 </td>2873 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2874 <p>6e-006</p>2875 </td>2876 </tr>2877 <tr style="height: 18.8pt;">2878 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2879 <p>rimA</p>2880 </td>2881 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2882 <p>Å</p>2883 </td>2884 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2885 <p>10</p>2886 </td>2887 </tr>2888 <tr style="height: 18.8pt;">2889 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2890 <p>rimB</p>2891 </td>2892 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2893 <p>Å</p>2894 </td>2895 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2896 <p>10</p>2897 </td>2898 </tr>2899 <tr style="height: 18.8pt;">2900 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2901 <p>rimC</p>2902 </td>2903 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2904 <p>Å</p>2905 </td>2906 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2907 <p>10</p>2908 </td>2909 </tr>2910 <tr style="height: 18.8pt;">2911 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2912 <p>longC</p>2913 </td>2914 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2915 <p>Å</p>2916 </td>2917 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2918 <p>400</p>2919 </td>2920 </tr>2921 <tr style="height: 18.8pt;">2922 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2923 <p>shortA</p>2924 </td>2925 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2926 <p>Å</p>2927 </td>2928 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2929 <p>35</p>2930 </td>2931 </tr>2932 <tr style="height: 18.8pt;">2933 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2934 <p>midB</p>2935 </td>2936 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2937 <p>Å</p>2938 </td>2939 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2940 <p>75</p>2941 </td>2942 </tr>2943 <tr style="height: 18.8pt;">2944 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143">2945 <p>scale</p>2946 </td>2947 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> </td>2948 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143">2949 <p>1</p>2950 </td>2951 </tr>2952 </tbody>2953 </table>2954 </div>2955 <p style="text-align: center;" align="center"><img id="Picture 33" src="img/image096.jpg" alt="" width="450" height="338" /></p>2956 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/256 data points).</b></p>2957 <p style="text-align: center;" align="center"><b> </b></p>2958 <p style="text-align: center;" align="center"><img id="Picture 34" src="img/image097.jpg" alt="" width="451" height="339" /></p>2959 <p style="text-align: center;" align="center"><b>Figure. 2D plot using the default values (w/(256X265) data points).</b></p>2960 2758 2961 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 2962 <p> REFERENCE</p> 2963 <p>see: Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 185-211.</p> 2964 <p>Equations (1), (13-14). (yes, it's in German) </p> 2965 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.23.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="EllipticalCylinderModel"></a><b><span style="font-size: 14pt;">Elliptical Cylinder Model</span></b></p> 2759 2760 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.21.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="EllipticalCylinderModel"></a><b><span style="font-size: 14pt;">Elliptical Cylinder Model</span></b></p> 2966 2761 <p>This function calculates the scattering from an oriented elliptical cylinder.</p> 2967 2762 <p><b>For 2D (orientated system):</b></p> … … 3092 2887 <p>REFERENCE</p> 3093 2888 <p style="text-indent: 0.25in;">L. A. Feigin and D. I. Svergun “Structure Analysis by Small-Angle X-Ray and Neutron Scattering”, Plenum, New York, (1987).</p> 3094 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 4.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="BarBellModel"></a><b><span style="font-size: 14pt;">BarBell(/DumBell)Model</span></b></p>2889 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.22.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="BarBellModel"></a><b><span style="font-size: 14pt;">BarBell(/DumBell)Model</span></b></p> 3095 2890 <p>Calculates the scattering from a barbell-shaped cylinder (This model simply becomes the DumBellModel when the length of the cylinder, L, is set to zero). That is, a sphereocylinder with spherical end caps that have a radius larger than that of the cylinder and the center of the end cap radius lies outside of the cylinder All dimensions of the barbell are considered to be monodisperse. See the diagram for the details of the geometry and restrictions on parameter values. </p> 3096 2891 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> … … 3223 3018 3224 3019 3225 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 5.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CappedCylinderModel"></a><b><span style="font-size: 14pt;">CappedCylinder(/ConvexLens)Model</span></b></p>3020 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.23.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CappedCylinderModel"></a><b><span style="font-size: 14pt;">CappedCylinder(/ConvexLens)Model</span></b></p> 3226 3021 <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> 3227 3022 <p style="margin-left: 0.85in; text-indent: -0.35in;"> </p> … … 3350 3145 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 3351 3146 3147 <!-- Pringle form factor AJJ 2014-02-17 --> 3148 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.24.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="PringleModel"></a><b><span style="font-size: 14pt;">PringleModel</span></b></p> 3149 <p>This model provides the form factor, P(<em>q</em>), for a 'pringle' or 'saddle-shaped' object (a hyperbolic paraboloid).</p> 3150 <p style="text-align: center;" align="center"> </p> 3151 <p>The returned value is in units of [cm-1], on absolute scale.</p> 3152 <p>The form factor calculated is:</p> 3153 <p style="text-align: center;" align="center"><img src="img/pringle_eqn_1.jpg" alt="" height='60px'/></p> 3154 <p>where</p> 3155 <p style="text-align: center;" align="center"><img src="img/pringle_eqn_2.jpg" alt="" height='120px'/></p> 3156 <p> </p> 3157 <p>The parameters of the model and a plot comparing the pringle model with the equivalent cylinder are shown below.</p> 3158 <div align="center"> 3159 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 3160 <tbody> 3161 <tr style="height: 18.8pt;"> 3162 <td style="border: 1pt solid width: 107pt; height: 18.8pt;" valign="top" width="143"> 3163 <p>Parameter name</p> 3164 </td> 3165 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3166 <p>Units</p> 3167 </td> 3168 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3169 <p>Default value</p> 3170 </td> 3171 </tr> 3172 <tr style="height: 18.8pt;"> 3173 <td style="border-width: medium 1pt 1pt; vertical-align: top; width: 107pt; height: 18.8pt;"> 3174 <p>background</p> 3175 </td> 3176 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3177 <p>cm-1</p> 3178 </td> 3179 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3180 <p>0.0</p> 3181 </td> 3182 </tr> 3183 <tr style="height: 18.8pt;"> 3184 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3185 <p>alpha</p> 3186 </td> 3187 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3188 <p></p> 3189 </td> 3190 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3191 <p>0.001</p> 3192 </td> 3193 </tr> 3194 <tr style="height: 18.8pt;"> 3195 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3196 <p>beta</p> 3197 </td> 3198 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3199 <p></p> 3200 </td> 3201 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3202 <p>0.02</p> 3203 </td> 3204 </tr> 3205 <tr style="height: 18.8pt;"> 3206 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3207 <p>radius</p> 3208 </td> 3209 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> </td> 3210 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3211 <p>60</p> 3212 </td> 3213 </tr> 3214 <tr style="height: 18.8pt;"> 3215 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3216 <p>scale</p> 3217 </td> 3218 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> </td> 3219 <p></p> 3220 </td> 3221 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3222 <p>1</p> 3223 </td> 3224 </tr> 3225 <tr style="height: 18.8pt;"> 3226 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3227 <p>sld_pringle</p> 3228 </td> 3229 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3230 <p>Å -2</p> 3231 </td> 3232 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3233 <p>1e-006</p> 3234 </td> 3235 </tr> 3236 <tr style="height: 18.8pt;"> 3237 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3238 <p>sld_solvent</p> 3239 </td> 3240 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3241 <p>Å -2</p> 3242 </td> 3243 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3244 <p>6.3e-006</p> 3245 </td> 3246 </tr> 3247 <tr style="height: 18.8pt;"> 3248 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3249 <p>thickness</p> 3250 </td> 3251 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3252 <p>Å</p> 3253 </td> 3254 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3255 <p>10</p> 3256 </td> 3257 </tr> 3258 </tbody> 3259 </table> 3260 </div> 3261 <p style="text-align: center;" align="center"><img id="Picture 492" src="img/pringle-vs-cylinder.png" alt="" width="455" height="351" /></p> 3262 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/150 data point).</b></p> 3263 <p>REFERENCE</p> 3264 <p>S. Alexandru Rautu, Private Communication.</p> 3352 3265 3353 3266 3354 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.26.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="EllipsoidModel"></a><b><span style="font-size: 14pt;">Ellipsoid Model</span></b></p> 3267 3268 3269 3270 3271 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.25.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="ParallelepipedModel"></a><b><span style="font-size: 14pt;">ParallelepipedModel (Magnetic 2D Model) </span></b></p> 3272 <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> 3273 For magnetic scattering, please see the '<a href="polar_mag_help.html">Polarization/Magnetic Scattering</a>' in Fitting Help. 3274 <p><span style="font-size: 14pt;"> </span></p> 3275 <p style="text-align: center;" align="center"><img src="img/image087.jpg" alt="" width="326" height="247" /></p> 3276 <p>The side of the solid must be satisfied the condition of A<B</p> 3277 <p>By this definition, assuming</p> 3278 <p>a = A/B<1; b=B/B=1; c=C/B>1, the form factor,</p> 3279 <p style="text-align: center;" align="center"><span style="position: relative; top: 48pt;"><img src="img/image088.PNG" alt="" /></span></p> 3280 <p>The contrast is defined as</p> 3281 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/image089.PNG" alt="" /></span></p> 3282 <p>The scattering intensity per unit volume is returned in the unit of [cm-1]; I(q) = <span style="font-family: Symbol;">f</span>P(q).</p> 3283 <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> 3284 <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> 3285 <p style="text-align: center;" align="center"><img src="img/image090.jpg"/></p> 3286 <p style="text-align: center;" align="center"><b>Figure. Definition of angles for 2D</b>.</p> 3287 <p style="text-align: center;" align="center"><img src="img/image091.jpg" alt="" width="379" height="256" /></p> 3288 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented pp against the detector plane.</p> 3289 <div align="center"> 3290 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 3291 <tbody> 3292 <tr style="height: 18.8pt;"> 3293 <td style="border: 1pt solid width: 107pt; height: 18.8pt;" valign="top" width="143"> 3294 <p>Parameter name</p> 3295 </td> 3296 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3297 <p>Units</p> 3298 </td> 3299 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3300 <p>Default value</p> 3301 </td> 3302 </tr> 3303 <tr style="height: 18.8pt;"> 3304 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3305 <p>background</p> 3306 </td> 3307 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3308 <p>cm-1</p> 3309 </td> 3310 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3311 <p>0.0</p> 3312 </td> 3313 </tr> 3314 <tr style="height: 18.8pt;"> 3315 <td style="border-width: medium 1pt 1pt; vertical-align: top; width: 107pt; height: 18.8pt;"> 3316 <p>contrast</p> 3317 </td> 3318 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3319 <p>Å -2</p> 3320 </td> 3321 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3322 <p>5e-006</p> 3323 </td> 3324 </tr> 3325 <tr style="height: 18.8pt;"> 3326 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3327 <p>long_c</p> 3328 </td> 3329 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3330 <p>Å</p> 3331 </td> 3332 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3333 <p>400</p> 3334 </td> 3335 </tr> 3336 <tr style="height: 18.8pt;"> 3337 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3338 <p>short_a</p> 3339 </td> 3340 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3341 <p>Å -2</p> 3342 </td> 3343 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3344 <p>35</p> 3345 </td> 3346 </tr> 3347 <tr style="height: 18.8pt;"> 3348 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3349 <p>short_b</p> 3350 </td> 3351 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3352 <p>Å</p> 3353 </td> 3354 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3355 <p>75</p> 3356 </td> 3357 </tr> 3358 <tr style="height: 18.8pt;"> 3359 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3360 <p>scale</p> 3361 </td> 3362 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> </td> 3363 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3364 <p>1</p> 3365 </td> 3366 </tr> 3367 </tbody> 3368 </table> 3369 </div> 3370 <p style="text-align: center;" align="center"><img id="Picture 492" src="img/image092.jpg" alt="" width="455" height="351" /></p> 3371 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/1000 data point).</b></p> 3372 <p style="margin-left: 1.35in; text-indent: -0.25in;"><span style="font-family: Symbol;">·</span><span style="font-size: 7pt;"> </span><b>Validation of the parallelepiped 2D model</b></p> 3373 <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> 3374 <p style="text-align: center;" align="center"><img id="Picture 104" src="img/image093.gif" alt="" width="481" height="299" /></p> 3375 <p style="text-align: center;" align="center"><b>Figure. Comparison between 1D and averaged 2D.</b></p> 3376 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 3377 <p>REFERENCE</p> 3378 <p>Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 185-211.</p> 3379 <p>Equations (1), (13-14). (in German)</p> 3380 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.26.</span></b><b><span style="font-size: 7pt;"> </span></b><b><span style="font-size: 14pt;"><a name="CSParallelepipedModel"></a>CSParallelepipedModel</span></b></p> 3381 <p>Calculates the form factor for a rectangular solid with a core-shell structure. The thickness and the scattering length density of the shell or "rim" can be different on all three (pairs) of faces. The form factor is normalized by the particle volume such that P(q) = scale*<f^2>/Vol + bkg, where < > is an average over all possible orientations of the rectangular solid. An instrument resolution smeared version is also provided.</p> 3382 <p>The function calculated is the form factor of the rectangular solid below. The core of the solid is defined by the dimensions ABC such that A < B < C. </p> 3383 <p style="text-align: center;" align="center"><img id="Picture 38" src="img/image087.jpg" alt="" width="326" height="247" /> </p> 3384 <p>There are rectangular "slabs" of thickness tA that add to the A dimension (on the BC faces). There are similar slabs on the AC (=tB) and AB (=tC) faces. The projection in the AB plane is then:</p> 3385 <p style="text-align: center;" align="center"><img src="img/image094.jpg" alt="" width="334" height="277" /></p> 3386 <p>The volume of the solid is:</p> 3387 <p><span style="position: relative; top: 5pt;"><img src="img/image095.PNG" alt="" /></span></p> 3388 <p>meaning that there are "gaps" at the corners of the solid.</p> 3389 <p>The intensity calculated follows the parallelepiped model, with the core-shell intensity being calculated as the square of the sum of the amplitudes of the core and shell, in the same manner as a core-shell sphere.</p> 3390 <p>For the calculation of the form factor to be valid, the sides of the solid MUST be chosen such that A < B < C. If this inequality in not satisfied, the model will not report an error, and the calculation will not be correct.</p> 3391 <p>FITTING NOTES:</p> 3392 <p>If the scale is set equal to the particle volume fraction, f, the returned value is the scattered intensity per unit volume, I(q) = f*P(q). However, no interparticle interference effects are included in this calculation.</p> 3393 <p>There are many parameters in this model. Hold as many fixed as possible with known values, or you will certainly end up at a solution that is unphysical.</p> 3394 <p>Constraints must be applied during fitting to ensure that the inequality A < B < C is not violated. The calculation will not report an error, but the results will not be correct.</p> 3395 <p>The returned value is in units of [cm-1], on absolute scale.</p> 3396 <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> 3397 <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> 3398 <p style="text-align: center;" align="center"><img id="Picture 102" src="img/image090.jpg" /></p> 3399 <p style="text-align: center;" align="center"><b>Figure. Definition of angles for 2D</b>.</p> 3400 <p style="text-align: center;" align="center"><img id="Picture 103" src="img/image091.jpg" alt="" width="379" height="256" /></p> 3401 <p style="text-align: center;" align="center">Figure. Examples of the angles for oriented cspp against the detector plane.</p> 3402 <p> TEST DATASET</p> 3403 <p>This example dataset is produced by running the Macro Plot_CSParallelepiped(), using 100 data points, qmin = 0.001 Å-1, qmax = 0.7 Å-1 and the below default values. </p> 3404 <div align="center"> 3405 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 3406 <tbody> 3407 <tr style="height: 18.8pt;"> 3408 <td style="border: 1pt solid width: 107pt; height: 18.8pt;" valign="top" width="143"> 3409 <p>Parameter name</p> 3410 </td> 3411 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3412 <p>Units</p> 3413 </td> 3414 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3415 <p>Default value</p> 3416 </td> 3417 </tr> 3418 <tr style="height: 18.8pt;"> 3419 <td style="border-width: medium 1pt 1pt; vertical-align: top; width: 107pt; height: 18.8pt;"> 3420 <p>background</p> 3421 </td> 3422 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3423 <p>cm-1</p> 3424 </td> 3425 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3426 <p>0.06</p> 3427 </td> 3428 </tr> 3429 <tr style="height: 18.8pt;"> 3430 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3431 <p>sld_pcore</p> 3432 </td> 3433 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3434 <p>Å -2</p> 3435 </td> 3436 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3437 <p>1e-006</p> 3438 </td> 3439 </tr> 3440 <tr style="height: 18.8pt;"> 3441 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3442 <p>sld_rimA</p> 3443 </td> 3444 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3445 <p>Å -2</p> 3446 </td> 3447 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3448 <p>2e-006</p> 3449 </td> 3450 </tr> 3451 <tr style="height: 18.8pt;"> 3452 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3453 <p>sld_rimB</p> 3454 </td> 3455 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3456 <p>Å -2</p> 3457 </td> 3458 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3459 <p>4e-006</p> 3460 </td> 3461 </tr> 3462 <tr style="height: 18.8pt;"> 3463 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3464 <p>sld_rimC</p> 3465 </td> 3466 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3467 <p>Å -2</p> 3468 </td> 3469 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3470 <p>2e-006</p> 3471 </td> 3472 </tr> 3473 <tr style="height: 18.8pt;"> 3474 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3475 <p>sld_solv</p> 3476 </td> 3477 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3478 <p>Å -2</p> 3479 </td> 3480 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3481 <p>6e-006</p> 3482 </td> 3483 </tr> 3484 <tr style="height: 18.8pt;"> 3485 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3486 <p>rimA</p> 3487 </td> 3488 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3489 <p>Å</p> 3490 </td> 3491 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3492 <p>10</p> 3493 </td> 3494 </tr> 3495 <tr style="height: 18.8pt;"> 3496 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3497 <p>rimB</p> 3498 </td> 3499 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3500 <p>Å</p> 3501 </td> 3502 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3503 <p>10</p> 3504 </td> 3505 </tr> 3506 <tr style="height: 18.8pt;"> 3507 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3508 <p>rimC</p> 3509 </td> 3510 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3511 <p>Å</p> 3512 </td> 3513 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3514 <p>10</p> 3515 </td> 3516 </tr> 3517 <tr style="height: 18.8pt;"> 3518 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3519 <p>longC</p> 3520 </td> 3521 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3522 <p>Å</p> 3523 </td> 3524 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3525 <p>400</p> 3526 </td> 3527 </tr> 3528 <tr style="height: 18.8pt;"> 3529 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3530 <p>shortA</p> 3531 </td> 3532 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3533 <p>Å</p> 3534 </td> 3535 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3536 <p>35</p> 3537 </td> 3538 </tr> 3539 <tr style="height: 18.8pt;"> 3540 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3541 <p>midB</p> 3542 </td> 3543 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3544 <p>Å</p> 3545 </td> 3546 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3547 <p>75</p> 3548 </td> 3549 </tr> 3550 <tr style="height: 18.8pt;"> 3551 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3552 <p>scale</p> 3553 </td> 3554 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> </td> 3555 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 3556 <p>1</p> 3557 </td> 3558 </tr> 3559 </tbody> 3560 </table> 3561 </div> 3562 <p style="text-align: center;" align="center"><img id="Picture 33" src="img/image096.jpg" alt="" width="450" height="338" /></p> 3563 <p style="text-align: center;" align="center"><b>Figure. 1D plot using the default values (w/256 data points).</b></p> 3564 <p style="text-align: center;" align="center"><b> </b></p> 3565 <p style="text-align: center;" align="center"><img id="Picture 34" src="img/image097.jpg" alt="" width="451" height="339" /></p> 3566 <p style="text-align: center;" align="center"><b>Figure. 2D plot using the default values (w/(256X265) data points).</b></p> 3567 3568 <p>Our model uses the form factor calculations implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006):</p> 3569 <p> REFERENCE</p> 3570 <p>see: Mittelbach and Porod, Acta Physica Austriaca 14 (1961) 185-211.</p> 3571 <p>Equations (1), (13-14). (yes, it's in German) </p> 3572 3573 3574 3575 3576 3577 3578 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.27.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="EllipsoidModel"></a><b><span style="font-size: 14pt;">Ellipsoid Model</span></b></p> 3355 3579 <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> 3356 3580 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> … … 3486 3710 <p><a name="_Ref173223004"></a>Figure 6: Comparison of the intensity for uniformly distributed ellipsoids calculated from our 2D model and the intensity from the NIST SANS analysis software. The parameters used were: Scale=1.0, Radius_a=20 Å, Radius_b=400 Å, Contrast=3e-6 Å -2, and Background=0.0 cm -1.</p> 3487 3711 <p> </p> 3488 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.2 7.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CoreShellEllipsoidModel"></a><b><span style="font-size: 14pt;">CoreShellEllipsoidModel </span></b></p>3712 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.28.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="CoreShellEllipsoidModel"></a><b><span style="font-size: 14pt;">CoreShellEllipsoidModel </span></b></p> 3489 3713 <p>This model provides the form factor, P(<em>q</em>), for a core shell ellipsoid (below) where the form factor is normalized by the volume of the cylinder. P(q) = scale*<f^2>/V+background where the volume V= 4pi/3*rmaj*rmin2 and the averaging < > is applied over all orientation for 1D. </p> 3490 3714 <p style="text-align: center;" align="center"> <img id="Picture 41" src="img/image125.gif" alt="" width="335" height="179" /></p> … … 3621 3845 <p>Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461.</p> 3622 3846 <p>Berr, S. J. Phys. Chem., 1987, 91, 4760.</p> 3623 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.28.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="TriaxialEllipsoidModel"></a><b><span style="font-size: 14pt;">TriaxialEllipsoidModel</span></b></p> 3847 3848 3849 3850 3851 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.29.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="TriaxialEllipsoidModel"></a><b><span style="font-size: 14pt;">TriaxialEllipsoidModel</span></b></p> 3624 3852 <p>This model provides the form factor, P(<em>q</em>), for an ellipsoid (below) where all three axes are of different lengths, i.e., Ra =< Rb =< Rc (Note that users should maintains this inequality for the all calculations). P(q) = scale*<f^2>/V+background where the volume V= 4pi/3*Ra*Rb*Rc, and the averaging < > is applied over all orientation for 1D. </p> 3625 3853 <p style="text-align: center;" align="center"> <img id="Picture 42" src="img/image128.jpg" alt="" width="376" height="226" /></p> … … 3736 3964 <p>REFERENCE</p> 3737 3965 <p>L. A. Feigin and D. I. Svergun “Structure Analysis by Small-Angle X-Ray and Neutron Scattering”, Plenum, New York, 1987.</p> 3738 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.29.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarModel"></a><b><span style="font-size: 14pt;">LamellarModel</span></b></p> 3966 3967 3968 3969 3970 3971 3972 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.30.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarModel"></a><b><span style="font-size: 14pt;">LamellarModel</span></b></p> 3739 3973 <p>This model provides the scattering intensity, I(<em>q</em>), for a lyotropic lamellar phase where a uniform SLD and random distribution in solution are assumed. The ploydispersion in the bilayer thickness can be applied from the GUI.</p> 3740 3974 <p>The scattering intensity I(q) is:</p> … … 3823 4057 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 3824 4058 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 3825 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 0.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarFFHGModel"></a><b><span style="font-size: 14pt;">LamellarFFHGModel</span></b></p>4059 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.31.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarFFHGModel"></a><b><span style="font-size: 14pt;">LamellarFFHGModel</span></b></p> 3826 4060 <p>This model provides the scattering intensity, I(<em>q</em>), for a lyotropic lamellar phase where a random distribution in solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail region.</p> 3827 4061 <p>The scattering intensity I(q) is:</p> … … 3932 4166 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 3933 4167 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 3934 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 1.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarPSModel"></a><b><span style="font-size: 14pt;">LamellarPSModel</span></b></p>4168 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.32.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarPSModel"></a><b><span style="font-size: 14pt;">LamellarPSModel</span></b></p> 3935 4169 <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> 3936 4170 <p>The scattering intensity I(q) is:</p> … … 4045 4279 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 4046 4280 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 4047 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 2.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarPSHGModel"></a><b><span style="font-size: 14pt;">LamellarPSHGModel</span></b></p>4281 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.33.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarPSHGModel"></a><b><span style="font-size: 14pt;">LamellarPSHGModel</span></b></p> 4048 4282 <p>This model provides the scattering intensity (<b>form factor</b> <b>*</b> <b>structure factor</b>), I(<em>q</em>), for a lyotropic lamellar phase where a random distribution in solution are assumed. The SLD of the head region is taken to be different from the SLD of the tail region.</p> 4049 4283 <p>The scattering intensity I(q) is:</p> … … 4192 4426 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 4193 4427 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 4194 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 3.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarPCrystalModel"></a><b><span style="font-size: 14pt;">LamellarPCrystalModel</span></b></p>4428 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.34.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarPCrystalModel"></a><b><span style="font-size: 14pt;">LamellarPCrystalModel</span></b></p> 4195 4429 <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> 4196 4430 <p>The scattering intensity I(q) is calculated as:</p> … … 4308 4542 <p>REFERENCE</p> 4309 4543 <p>M. Bergstrom, J. S. Pedersen, P. Schurtenberger, S. U. Egelhaaf, J. Phys. Chem. B, 103 (1999) 9888-9897.</p> 4310 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 4.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="SCCrystalModel"></a><b><span style="font-size: 14pt;">SC(Simple Cubic Para-)CrystalModel</span></b></p>4544 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.35.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="SCCrystalModel"></a><b><span style="font-size: 14pt;">SC(Simple Cubic Para-)CrystalModel</span></b></p> 4311 4545 <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> 4312 4546 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 4432 4666 <p style="text-align: center;" align="center"><b><img src="img/image157.jpg" alt="" width="447" height="322" /></b></p> 4433 4667 <p style="text-align: center;" align="center"><b>Figure. 2D plot using the default values (w/200X200 pixels).</b></p> 4434 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 5.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="FCCrystalModel"></a><b><span style="font-size: 14pt;">FC(Face Centered Cubic Para-)CrystalModel</span></b></p>4668 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.36.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="FCCrystalModel"></a><b><span style="font-size: 14pt;">FC(Face Centered Cubic Para-)CrystalModel</span></b></p> 4435 4669 <p>Calculates the scattering from a face-centered cubic lattice with paracrystalline distortion. Thermal vibrations are considered to be negligible, and the size of the paracrystal is infinitely large. Paracrystalline distortion is assumed to be isotropic and characterized by a Gaussian distribution. </p> 4436 4670 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 4557 4791 <p style="text-align: center;" align="center"><img src="img/image166.jpg" alt="" width="473" height="352" /></p> 4558 4792 <p style="text-align: center;" align="center"><b>Figure. 2D plot using the default values (w/200X200 pixels).</b></p> 4559 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.3 6.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="BCCrystalModel"></a><b><span style="font-size: 14pt;">BC(Body Centered Cubic Para-)CrystalModel</span></b></p>4793 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.37.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="BCCrystalModel"></a><b><span style="font-size: 14pt;">BC(Body Centered Cubic Para-)CrystalModel</span></b></p> 4560 4794 <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> 4561 4795 <p>The scattering intensity I(q) is calculated as:</p>
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