Changeset a110e37f in sasview
- Timestamp:
- Apr 27, 2014 3:12:49 PM (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:
- 9611dd6
- Parents:
- 3600c12
- Location:
- src/sans/models/media
- Files:
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- 13 added
- 1 edited
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model_functions.html (modified) (12 diffs)
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src/sans/models/media/model_functions.html
ra857c0a ra110e37f 8 8 <li><a href="#Introduction"><b>Introduction</b></a></li> 9 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> 59 </li> 60 </ul> 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 <a href="#RectangularHollowPrismInfThinWallsModel">RectangularHollowPrismInfThinWallsModel</a>, 43 <a href="#RectangularPrismModel">RectangularPrismModel</a>, 44 <a href="#RectangularHollowPrismModel">RectangularHollowPrismModel</a> 45 </li> 46 <li>Ellipsoids:<br/> 47 <a href="#EllipsoidModel">EllipsoidModel</a>, 48 <a href="#CoreShellEllipsoidModel">CoreShellEllipsoidModel</a>, 49 <a href="#CoreShellEllipsoidXTModel">CoreShellEllipsoidXTModel</a>, 50 <a href="#TriaxialEllipsoidModel">TriaxialEllipsoidModel</a> 51 </li> 52 <li>Lamellar:<br/> 53 <a href="#LamellarModel">LamellarModel</a>, 54 <a href="#LamellarFFHGModel">LamellarFFHGModel</a>, 55 <a href="#LamellarPSModel">LamellarPSModel</a>, 56 <a href="#LamellarPSHGModel">LamellarPSHGModel</a> 57 </li> 58 <li>Paracrystals:<br/> 59 <a href="#LamellarPCrystalModel">LamellarPCrystalModel</a>, 60 <a href="#SCCrystalModel">SCCrystalModel</a>, 61 <a href="#FCCrystalModel">FCCrystalModel</a>, 62 <a href="#BCCrystalModel">BCCrystalModel</a> 63 </li> 64 </ul> 61 65 <li><a href="#Shape-Independent"><b>Shape-Independent</b></a>: 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 66 <a href="#Absolute%20Power_Law">AbsolutePower_Law</a>, 67 <a href="#BEPolyelectrolyte">BEPolyelectrolyte</a>, 68 <a href="#BroadPeakModel">BroadPeakModel</a>, 69 <a href="#CorrLength">CorrLength</a>, 70 <a href="#DABModel">DABModel</a>, 71 <a href="#Debye">Debye</a>, 72 <a href="#Number_Density_Fractal">FractalModel</a>, 73 <a href="#FractalCoreShell">FractalCoreShell</a>, 74 <a href="#GaussLorentzGel">GaussLorentzGel</a>, 75 <a href="#Guinier">Guinier</a>, 76 <a href="#GuinierPorod">GuinierPorod</a>, 77 <a href="#Lorentz">Lorentz</a>, 78 <a href="#Mass_Fractal">MassFractalModel</a>, 79 <a href="#MassSurface_Fractal">MassSurfaceFractal</a>, 80 <a href="#Peak%20Gauss%20Model">PeakGaussModel</a>, 81 <a href="#Peak%20Lorentz%20Model">PeakLorentzModel</a>, 82 <a href="#Poly_GaussCoil">Poly_GaussCoil</a>, 83 <a href="#PolymerExclVolume">PolyExclVolume</a>, 84 <a href="#PorodModel">PorodModel</a>, 85 <a href="#RPA10Model">RPA10Model</a>, 86 <a href="#StarPolymer">StarPolymer</a>, 87 <a href="#Surface_Fractal">SurfaceFractalModel</a>, 88 <a href="#TeubnerStreyModel">Teubner Strey</a>, 89 <a href="#TwoLorentzian">TwoLorentzian</a>, 90 <a href="#TwoPowerLaw">TwoPowerLaw</a>, 91 <a href="#UnifiedPowerRg">UnifiedPowerRg</a>, 92 <a href="#LineModel">LineModel</a>, 93 <a href="#ReflectivityModel">ReflectivityModel</a>, 94 <a href="#ReflectivityIIModel">ReflectivityIIModel</a>, 95 <a href="#GelFitModel">GelFitModel</a>.</li> 96 93 97 <li><a href="#Model"><b>Customized Models</b></a>: 94 95 96 97 98 99 100 98 <a href="#testmodel">testmodel</a>, 99 <a href="#testmodel_2">testmodel_2</a>, 100 <a href="#sum_p1_p2">sum_p1_p2</a>, 101 <a href="#sum_Ap1_1_Ap2">sum_Ap1_1_Ap2</a>, 102 <a href="#polynomial5">polynomial5</a>, 103 <a href="#sph_bessel_jn">sph_bessel_jn</a>.</li> 104 101 105 <li><a href="#Structure_Factors"><b>Structure Factors</b></a>: 102 103 104 105 106 106 <a href="#HardsphereStructure">HardSphereStructure</a>, 107 <a href="#SquareWellStructure">SquareWellStructure</a>, 108 <a href="#HayterMSAStructure">HayterMSAStructure</a>, 109 <a href="#StickyHSStructure">StickyHSStructure</a>.</li> 110 107 111 <li><a href="#References"><b>References</b></a></li> 108 112 </ul> … … 3575 3579 3576 3580 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> 3581 <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><b><span style="font-size: 14pt;"><a name="RectangularHollowPrismInfThinWallsModel"></a>RectangularHollowPrismInfThinWallsModel</span></b></p> 3582 3583 <p>This model provides the form factor, P( <em>q</em>), for a hollow rectangular prism 3584 with infinitely thin walls.</p> 3585 <p><em>Definition</em></p> 3586 <p>The 1D scattering intensity for this model is calculated according to the equations given by 3587 Nayuk and Huber (Nayuk, 2012).</p> 3588 <p>Assuming a hollow parallelepiped with infinitely thin walls, edge lengths <span class="formula"><i>A</i>â 3589 â€â 3590 <i>B</i>â 3591 â€â 3592 <i>C</i></span> 3593 3594 and presenting an orientation with respect to the scattering vector given by Ξ and Ï, 3595 where Ξ is the angle between the <em>z</em> axis and the longest axis of the parallelepiped <em>C</em>, and Ï 3596 is the angle between the scattering vector (lying in the <em>xy</em> plane) and the <em>y</em> axis, 3597 the form factor is given by:</p> 3598 3599 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrismInfThinWalls_1.png" alt="" /></span></p> 3600 3601 <p>where</p> 3602 3603 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrismInfThinWalls_2.png" alt="" /></span></p> 3604 3605 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrismInfThinWalls_3.png" alt="" /></span></p> 3606 3607 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrismInfThinWalls_4.png" alt="" /></span></p> 3608 3609 <p>and</p> 3610 3611 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrismInfThinWalls_5.png" alt="" /></span></p> 3612 3613 <p>The 1D scattering intensity is calculated as:</p> 3614 3615 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrismInfThinWalls_6.png" alt="" /></span></p> 3616 3617 <p>where <em>V</em> is the volume of the rectangular prism, <span class="formula"><i>Ï</i><sub><span class="mbox">pipe</span></sub></span> 3618 is the scattering length of the 3619 parallelepiped, <span class="formula"><i>Ï</i><sub><span class="mbox">solvent</span></sub></span> 3620 is the scattering length of the solvent, and 3621 (if the data are in absolute scale) scale represents the volume fraction (which is unitless) .</p> 3622 <p>The 2D scattering intensity is not computed by this model.</p> 3623 <p>The returned value is scaled to units of cm<sup>-1</sup> and the parameters of the RectangularHollowPrismInfThinWallModel 3624 are the following:</p> 3625 <table border="1" class="docutils"> 3626 <colgroup> 3627 <col width="40%" /> 3628 <col width="23%" /> 3629 <col width="37%" /> 3630 </colgroup> 3631 <thead valign="bottom"> 3632 <tr><th class="head">Parameter name</th> 3633 <th class="head">Units</th> 3634 <th class="head">Default value</th> 3635 </tr> 3636 </thead> 3637 <tbody valign="top"> 3638 <tr><td>scale</td> 3639 <td>None</td> 3640 <td>1</td> 3641 </tr> 3642 <tr><td>short_side</td> 3643 <td>â«</td> 3644 <td>35</td> 3645 </tr> 3646 <tr><td>b2a_ratio</td> 3647 <td>None</td> 3648 <td>1</td> 3649 </tr> 3650 <tr><td>c2a_ratio</td> 3651 <td>None</td> 3652 <td>1</td> 3653 </tr> 3654 <tr><td>sldPipe</td> 3655 <td>â«<sup>-2</sup></td> 3656 <td>6.3e-6</td> 3657 </tr> 3658 <tr><td>sldSolv</td> 3659 <td>â«<sup>-2</sup></td> 3660 <td>1.0e-6</td> 3661 </tr> 3662 <tr><td>background</td> 3663 <td>cm<sup>-1</sup></td> 3664 <td>0</td> 3665 </tr> 3666 </tbody> 3667 </table> 3668 <p>REFERENCES</p> 3669 <ol class="upperalpha simple" start="18"> 3670 <li>Nayuk and K. Huber, <em>Z. Phys. Chem.</em> 226 (2012) 837-854.</li> 3671 </ol> 3672 <p><em>Validation of the RectangularHollowPrismInfThinWallsModel</em></p> 3673 <p>Validation of the code was done qualitatively by comparing the output of the 1D model to the curves 3674 shown in (Nayuk, 2012).</p> 3675 3676 3677 3678 3679 3680 3681 3682 3683 3684 3685 3686 3687 <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><b><span style="font-size: 14pt;"><a name="RectangularPrismModel"></a>RectangularPrismModel</span></b></p> 3688 <p>This model provides the form factor, P( <em>q</em>), for a rectangular prism.</p> 3689 <p>Note that this model is almost totally equivalent to the existing 3690 ParallelepipedModel. The only difference is that the way the 3691 relevant parameters are defined here (<em>a</em>, <em>b/a</em>, <em>c/a</em> instead of <em>a</em>, <em>b</em>, <em>c</em>) 3692 allows to use polydispersity with this model while keeping the shape 3693 of the prism (e.g. setting <em>b/a</em> = 1 and <em>c/a</em> = 1 and applying polydispersity 3694 to <em>a</em> will generate a distribution of cubes of different sizes).</p> 3695 <p><em>Definition</em></p> 3696 <p>The 1D scattering intensity for this model was calculated by Mittelbach and Porod (Mittelbach, 1961), 3697 but the implementation here is closer to the equations given by Nayuk and Huber (Nayuk, 2012).</p> 3698 <p>The scattering from a massive parallelepiped with an orientation with respect to the scattering vector 3699 given by Ξ and Ï is given by:</p> 3700 3701 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularPrism_1.png" alt="" /></span></p> 3702 3703 <p>where <em>A</em>, <em>B</em> and <em>C</em> are the sides of the parallelepiped and must fulfill <span class="formula"><i>A</i>â 3704 â€â 3705 <i>B</i>â 3706 â€â 3707 <i>C</i></span> 3708 , 3709 Ξ is the angle between the <em>z</em> axis and the longest axis of the parallelepiped <em>C</em>, and Ï 3710 is the angle between the scattering vector (lying in the <em>xy</em> plane) and the <em>y</em> axis.</p> 3711 <p>The normalized form factor in 1D is obtained averaging over all possible orientations:</p> 3712 3713 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularPrism_2.png" alt="" /></span></p> 3714 3715 <p>The 1D scattering intensity is calculated as:</p> 3716 3717 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularPrism_3.png" alt="" /></span></p> 3718 3719 <p>where <em>V</em> is the volume of the rectangular prism, <span class="formula"><i>Ï</i><sub><span class="mbox">pipe</span></sub></span> 3720 is the scattering length of the 3721 parallelepiped, <span class="formula"><i>Ï</i><sub><span class="mbox">solvent</span></sub></span> 3722 is the scattering length of the solvent, and 3723 (if the data are in absolute scale) scale represents the volume fraction (which is unitless) .</p> 3724 <p>The 2D scattering intensity is not computed by this model.</p> 3725 <p>The returned value is scaled to units of cm<sup>-1</sup> and the parameters of the RectangularPrismModel are the following:</p> 3726 <table border="1" class="docutils"> 3727 <colgroup> 3728 <col width="40%" /> 3729 <col width="23%" /> 3730 <col width="37%" /> 3731 </colgroup> 3732 <thead valign="bottom"> 3733 <tr><th class="head">Parameter name</th> 3734 <th class="head">Units</th> 3735 <th class="head">Default value</th> 3736 </tr> 3737 </thead> 3738 <tbody valign="top"> 3739 <tr><td>scale</td> 3740 <td>None</td> 3741 <td>1</td> 3742 </tr> 3743 <tr><td>short_side</td> 3744 <td>â«</td> 3745 <td>35</td> 3746 </tr> 3747 <tr><td>b2a_ratio</td> 3748 <td>None</td> 3749 <td>1</td> 3750 </tr> 3751 <tr><td>c2a_ratio</td> 3752 <td>None</td> 3753 <td>1</td> 3754 </tr> 3755 <tr><td>sldPipe</td> 3756 <td>â«<sup>-2</sup></td> 3757 <td>6.3e-6</td> 3758 </tr> 3759 <tr><td>sldSolv</td> 3760 <td>â«<sup>-2</sup></td> 3761 <td>1.0e-6</td> 3762 </tr> 3763 <tr><td>background</td> 3764 <td>cm<sup>-1</sup></td> 3765 <td>0</td> 3766 </tr> 3767 </tbody> 3768 </table> 3769 <p>REFERENCES</p> 3770 <ol class="upperalpha simple" start="16"> 3771 <li>Mittelbach and G. Porod, <em>Acta Physica Austriaca</em> 14 (1961) 185-211.</li> 3772 </ol> 3773 <ol class="upperalpha simple" start="18"> 3774 <li>Nayuk and K. Huber, <em>Z. Phys. Chem.</em> 226 (2012) 837-854.</li> 3775 </ol> 3776 <p><em>Validation of the RectangularPrismModel</em></p> 3777 <p>Validation of the code was done by comparing the output of the 1D model to the output of the existing 3778 parallelepiped model.</p> 3779 3780 3781 3782 3783 3784 3785 3786 3787 3788 3789 <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><b><span style="font-size: 14pt;"><a name="RectangularHollowPrismModel"></a>RectangularHollowPrismModel</span></b></p> 3790 <p>This model provides the form factor, P( <em>q</em>), for a hollow rectangular parallelepiped 3791 with a wall thickness Î.</p> 3792 <p><em>Definition</em></p> 3793 <p>The 1D scattering intensity for this model is calculated by forming the difference of the 3794 amplitudes of two massive parallelepipeds differing in their outermost dimensions in 3795 each direction by the same length increment 2 Î (Nayuk, 2012).</p> 3796 <p>As in the case of the massive parallelepiped, the scattering amplitude is computed for a particular 3797 orientation of the parallelepiped with respect to the scattering vector and then averaged over all 3798 possible orientations, giving:</p> 3799 3800 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrism_1.png" alt="" /></span></p> 3801 3802 <p>where Ξ is the angle between the <em>z</em> axis and the longest axis of the parallelepiped, Ï 3803 is the angle between the scattering vector (lying in the <em>xy</em> plane) and the <em>y</em> axis, and:</p> 3804 3805 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrism_2.png" alt="" /></span></p> 3806 3807 <p>where <em>A</em>, <em>B</em> and <em>C</em> are the external sides of the parallelepiped fulfilling <span class="formula"><i>A</i>â 3808 â€â 3809 <i>B</i>â 3810 â€â 3811 <i>C</i></span> 3812 , 3813 and the volume <em>V</em> of the parallelepiped is:</p> 3814 3815 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrism_3.png" alt="" /></span></p> 3816 3817 <p>The 1D scattering intensity is calculated as:</p> 3818 3819 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrism_4.png" alt="" /></span></p> 3820 3821 <p>where <span class="formula"><i>Ï</i><sub><span class="mbox">pipe</span></sub></span> 3822 is the scattering length of the 3823 parallelepiped, <span class="formula"><i>Ï</i><sub><span class="mbox">solvent</span></sub></span> 3824 is the scattering length of the solvent, and 3825 (if the data are in absolute scale) scale represents the volume fraction (which is unitless) .</p> 3826 <p>The 2D scattering intensity is not computed by this model.</p> 3827 <p>The returned value is scaled to units of cm<sup>-1</sup> and the parameters of the RectangularHollowPrismModel 3828 are the following:</p> 3829 <table border="1" class="docutils"> 3830 <colgroup> 3831 <col width="40%" /> 3832 <col width="23%" /> 3833 <col width="37%" /> 3834 </colgroup> 3835 <thead valign="bottom"> 3836 <tr><th class="head">Parameter name</th> 3837 <th class="head">Units</th> 3838 <th class="head">Default value</th> 3839 </tr> 3840 </thead> 3841 <tbody valign="top"> 3842 <tr><td>scale</td> 3843 <td>None</td> 3844 <td>1</td> 3845 </tr> 3846 <tr><td>short_side</td> 3847 <td>â«</td> 3848 <td>35</td> 3849 </tr> 3850 <tr><td>b2a_ratio</td> 3851 <td>None</td> 3852 <td>1</td> 3853 </tr> 3854 <tr><td>c2a_ratio</td> 3855 <td>None</td> 3856 <td>1</td> 3857 </tr> 3858 <tr><td>thickness</td> 3859 <td>â«</td> 3860 <td>1</td> 3861 </tr> 3862 <tr><td>sldPipe</td> 3863 <td>â«<sup>-2</sup></td> 3864 <td>6.3e-6</td> 3865 </tr> 3866 <tr><td>sldSolv</td> 3867 <td>â«<sup>-2</sup></td> 3868 <td>1.0e-6</td> 3869 </tr> 3870 <tr><td>background</td> 3871 <td>cm<sup>-1</sup></td> 3872 <td>0</td> 3873 </tr> 3874 </tbody> 3875 </table> 3876 <p>REFERENCES</p> 3877 <ol class="upperalpha simple" start="18"> 3878 <li>Nayuk and K. Huber, <em>Z. Phys. Chem.</em> 226 (2012) 837-854.</li> 3879 </ol> 3880 <p><em>Validation of the RectangularHollowPrismModel</em></p> 3881 <p>Validation of the code was done qualitatively by comparing the output of the 1D model to the curves 3882 shown in (Nayuk, 2012).</p> 3883 3884 3885 3886 3887 3888 3889 3890 3891 3892 3893 3894 3895 3896 3897 3898 3899 3900 3901 <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="EllipsoidModel"></a><b><span style="font-size: 14pt;">Ellipsoid Model</span></b></p> 3579 3902 <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> 3580 3903 <p style="margin-left: 0.85in; text-indent: -0.35in;"><b>1.1.</b><b><span style="font-size: 7pt;"> </span>Definition</b></p> … … 3710 4033 <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> 3711 4034 <p> </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>4035 <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="CoreShellEllipsoidModel"></a><b><span style="font-size: 14pt;">CoreShellEllipsoidModel </span></b></p> 3713 4036 <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> 3714 4037 <p style="text-align: center;" align="center"> <img id="Picture 41" src="img/image125.gif" alt="" width="335" height="179" /></p> … … 3849 4172 3850 4173 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> 4174 4175 4176 4177 4178 4179 4180 <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="CoreShellEllipsoidXTModel"></a><b><span style="font-size: 14pt;">CoreShellEllipsoidXTModel </span></b></p> 4181 4182 <p>An alternative version of P( *q*) for the core plus shell ellipsoid (see 4183 CoreShellEllipsoidModel), having as 4184 parameters the core axial ratio X and a shell thickness, which are more often 4185 what we would like to determine and behave better when polydispersity is 4186 applied than the four independent radii in the original model.</p> 4187 4188 <p>The geometric parameters are: equat_core = equatorial core radius = Rminor_core, 4189 X_core = polar_core/equat_core = Rmajor_core/Rminor_core 4190 T_shell = equat_outer - equat_core = Rminor_outer - Rminor_core, 4191 XpolarShell = Tpolar_shell/T_shell = (Rmajor_outer - Rmajor_core)/(Rminor_outer - Rminor_core)</p> 4192 4193 <p>In terms of the original radii: 4194 polar_core = equat_core * X_core, equat_shell = equat_core + T_shell 4195 polar_shell = equat_core * X_core + T_shell*XpolarShell 4196 (where we note that "shell" perhaps confusingly, relates to the outer radius).</p> 4197 4198 <p>When X_core < 1 the core is oblate, when X_core > 1 it is prolate, 4199 X_core =1 is a spherical core. For a fixed shell thickness XpolarShell = 1, to scale the 4200 shell thickness pro-rata with the radius XpolarShell = X_core.</p> 4201 4202 <p>When including and S(Q), the radius in S(Q) is calculated to be that of a 4203 sphere with the same 2nd virial coefficient of the outr surface of the ellipsoid. 4204 This may have some undesirable effects if the aspect ratio of the ellipsoid is 4205 large ( X<<1 or X>>1), when the S(Q) which assumes spheres wil not in any case be valid.</p> 4206 4207 <p>If SANS data are in absolute units, and sld's are correct, then "scale" should be the total 4208 volume fraction of the "outer particle". When S(Q) is introduced this moves to the S(Q) 4209 volume fraction, and "scale" should then be 1.0, or contain some ther units conversion factor 4210 if you have say Xray data.</p> 4211 4212 4213 <div align="center"> 4214 <table style="border-collapse: collapse;" border="2" cellspacing="0" cellpadding="0"> 4215 <tbody> 4216 <tr style="height: 18.8pt;"> 4217 <td style="border: 1pt solid width: 107pt; height: 18.8pt;" valign="top" width="143"> 4218 <p>Parameter name</p> 4219 </td> 4220 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4221 <p>Units</p> 4222 </td> 4223 <td style="border-width: 1pt 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4224 <p>Default value</p> 4225 </td> 4226 </tr> 4227 <tr style="height: 18.8pt;"> 4228 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4229 <p>background</p> 4230 </td> 4231 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4232 <p>cm-1</p> 4233 </td> 4234 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4235 <p>0.001</p> 4236 </td> 4237 </tr> 4238 <tr style="height: 18.8pt;"> 4239 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4240 <p>equat_core</p> 4241 </td> 4242 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4243 <p>Å</p> 4244 </td> 4245 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4246 <p>20</p> 4247 </td> 4248 </tr> 4249 <tr style="height: 18.8pt;"> 4250 <td style="border-width: medium 1pt 1pt; vertical-align: top; width: 107pt; height: 18.8pt;"> 4251 <p>scale</p> 4252 </td> 4253 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4254 <p></p> 4255 </td> 4256 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4257 <p>0.05</p> 4258 </td> 4259 </tr> 4260 <tr style="height: 18.8pt;"> 4261 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4262 <p>sld_core</p> 4263 </td> 4264 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4265 <p>Å -2</p> 4266 </td> 4267 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4268 <p>2.0e-6</p> 4269 </td> 4270 </tr> 4271 <tr style="height: 18.8pt;"> 4272 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4273 <p>sld_shell</p> 4274 </td> 4275 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4276 <p>Å -2</p> 4277 </td> 4278 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4279 <p>1.0e-6</p> 4280 </td> 4281 </tr> 4282 <tr style="height: 18.8pt;"> 4283 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4284 <p>sld_solv</p> 4285 </td> 4286 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4287 <p>Å -2</p> 4288 </td> 4289 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4290 <p>6.3e-6</p> 4291 </td> 4292 </tr> 4293 <tr style="height: 18.8pt;"> 4294 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4295 <p>T_shell</p> 4296 </td> 4297 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4298 <p>Å</p> 4299 </td> 4300 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4301 <p>30</p> 4302 </td> 4303 </tr> 4304 <tr style="height: 18.8pt;"> 4305 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4306 <p>X_core</p> 4307 </td> 4308 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4309 <p></p> 4310 </td> 4311 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4312 <p>3.0</p> 4313 </td> 4314 </tr> 4315 <tr style="height: 18.8pt;"> 4316 <td style="border-width: medium 1pt 1pt; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4317 <p>XpolarShell</p> 4318 </td> 4319 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"></td> 4320 <td style="border-width: medium 1pt 1pt medium; width: 107pt; height: 18.8pt;" valign="top" width="143"> 4321 <p>1.0</p> 4322 </td> 4323 </tr> 4324 </tbody> 4325 </table> 4326 </div> 4327 4328 4329 4330 4331 4332 4333 4334 4335 4336 <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="TriaxialEllipsoidModel"></a><b><span style="font-size: 14pt;">TriaxialEllipsoidModel</span></b></p> 3852 4337 <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> 3853 4338 <p style="text-align: center;" align="center"> <img id="Picture 42" src="img/image128.jpg" alt="" width="376" height="226" /></p> … … 3970 4455 3971 4456 3972 <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="LamellarModel"></a><b><span style="font-size: 14pt;">LamellarModel</span></b></p>4457 <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="LamellarModel"></a><b><span style="font-size: 14pt;">LamellarModel</span></b></p> 3973 4458 <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> 3974 4459 <p>The scattering intensity I(q) is:</p> … … 4057 4542 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 4058 4543 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 4059 <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="LamellarFFHGModel"></a><b><span style="font-size: 14pt;">LamellarFFHGModel</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="LamellarFFHGModel"></a><b><span style="font-size: 14pt;">LamellarFFHGModel</span></b></p> 4060 4545 <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> 4061 4546 <p>The scattering intensity I(q) is:</p> … … 4166 4651 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 4167 4652 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 4168 <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="LamellarPSModel"></a><b><span style="font-size: 14pt;">LamellarPSModel</span></b></p>4653 <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="LamellarPSModel"></a><b><span style="font-size: 14pt;">LamellarPSModel</span></b></p> 4169 4654 <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> 4170 4655 <p>The scattering intensity I(q) is:</p> … … 4279 4764 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 4280 4765 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 4281 <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="LamellarPSHGModel"></a><b><span style="font-size: 14pt;">LamellarPSHGModel</span></b></p>4766 <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="LamellarPSHGModel"></a><b><span style="font-size: 14pt;">LamellarPSHGModel</span></b></p> 4282 4767 <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> 4283 4768 <p>The scattering intensity I(q) is:</p> … … 4426 4911 <p>Nallet, Laversanne, and Roux, J. Phys. II France, 3, (1993) 487-502.</p> 4427 4912 <p> also in J. Phys. Chem. B, 105, (2001) 11081-11088.</p> 4428 <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="LamellarPCrystalModel"></a><b><span style="font-size: 14pt;">LamellarPCrystalModel</span></b></p>4913 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.38.</span></b><b><span style="font-size: 7pt;"> </span></b><a name="LamellarPCrystalModel"></a><b><span style="font-size: 14pt;">LamellarPCrystalModel</span></b></p> 4429 4914 <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> 4430 4915 <p>The scattering intensity I(q) is calculated as:</p> … … 4542 5027 <p>REFERENCE</p> 4543 5028 <p>M. Bergstrom, J. S. Pedersen, P. Schurtenberger, S. U. Egelhaaf, J. Phys. Chem. B, 103 (1999) 9888-9897.</p> 4544 <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="SCCrystalModel"></a><b><span style="font-size: 14pt;">SC(Simple Cubic Para-)CrystalModel</span></b></p>5029 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.39.</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> 4545 5030 <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> 4546 5031 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 4666 5151 <p style="text-align: center;" align="center"><b><img src="img/image157.jpg" alt="" width="447" height="322" /></b></p> 4667 5152 <p style="text-align: center;" align="center"><b>Figure. 2D plot using the default values (w/200X200 pixels).</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>5153 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.40.</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> 4669 5154 <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> 4670 5155 <p>The returned value is scaled to units of [cm-1sr-1], absolute scale.</p> … … 4791 5276 <p style="text-align: center;" align="center"><img src="img/image166.jpg" alt="" width="473" height="352" /></p> 4792 5277 <p style="text-align: center;" align="center"><b>Figure. 2D plot using the default values (w/200X200 pixels).</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>5278 <p style="margin-left: 0.55in; text-indent: -0.3in;"><b><span style="font-size: 14pt;">2.41.</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> 4794 5279 <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> 4795 5280 <p>The scattering intensity I(q) is calculated as:</p>
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