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- Apr 27, 2014 3:33:24 PM (11 years ago)
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src/sans/models/media/model_functions.html
ra110e37f r9611dd6 3586 3586 <p>The 1D scattering intensity for this model is calculated according to the equations given by 3587 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 3588 <p>Assuming a hollow parallelepiped with infinitely thin walls, edge lengths A ≤ B ≤ C 3589 <span class="formula"><i>A</i>â 3589 3590 â€â 3590 3591 <i>B</i>â … … 3592 3593 <i>C</i></span> 3593 3594 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 Ï3595 and presenting an orientation with respect to the scattering vector given by θ and φ, 3596 where θ is the angle between the <em>z</em> axis and the longest axis of the parallelepiped <em>C</em>, and φ 3596 3597 is the angle between the scattering vector (lying in the <em>xy</em> plane) and the <em>y</em> axis, 3597 3598 the form factor is given by:</p> … … 3615 3616 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrismInfThinWalls_6.png" alt="" /></span></p> 3616 3617 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 <p>where <em>V</em> is the volume of the rectangular prism, <span class="formula"><i>ρ</i><sub><span class="mbox">pipe</span></sub></span> 3618 3619 is the scattering length of the 3619 parallelepiped, <span class="formula"><i> Ï</i><sub><span class="mbox">solvent</span></sub></span>3620 parallelepiped, <span class="formula"><i>ρ</i><sub><span class="mbox">solvent</span></sub></span> 3620 3621 is the scattering length of the solvent, and 3621 3622 (if the data are in absolute scale) scale represents the volume fraction (which is unitless) .</p> … … 3641 3642 </tr> 3642 3643 <tr><td>short_side</td> 3643 <td> â«</td>3644 <td>Å</td> 3644 3645 <td>35</td> 3645 3646 </tr> … … 3653 3654 </tr> 3654 3655 <tr><td>sldPipe</td> 3655 <td> â«<sup>-2</sup></td>3656 <td>Å<sup>-2</sup></td> 3656 3657 <td>6.3e-6</td> 3657 3658 </tr> 3658 3659 <tr><td>sldSolv</td> 3659 <td> â«<sup>-2</sup></td>3660 <td>Å<sup>-2</sup></td> 3660 3661 <td>1.0e-6</td> 3661 3662 </tr> … … 3697 3698 but the implementation here is closer to the equations given by Nayuk and Huber (Nayuk, 2012).</p> 3698 3699 <p>The scattering from a massive parallelepiped with an orientation with respect to the scattering vector 3699 given by Ξ and Ïis given by:</p>3700 given by θ and φ is given by:</p> 3700 3701 3701 3702 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularPrism_1.png" alt="" /></span></p> 3702 3703 3703 3704 <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€â3705 ≤â 3705 3706 <i>B</i>â 3706 â€â3707 ≤â 3707 3708 <i>C</i></span> 3708 3709 , 3709 Ξ is the angle between the <em>z</em> axis and the longest axis of the parallelepiped <em>C</em>, and Ï 3710 θ is the angle between the <em>z</em> axis and the longest axis of the parallelepiped <em>C</em>, and φ 3710 3711 is the angle between the scattering vector (lying in the <em>xy</em> plane) and the <em>y</em> axis.</p> 3711 3712 <p>The normalized form factor in 1D is obtained averaging over all possible orientations:</p> … … 3717 3718 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularPrism_3.png" alt="" /></span></p> 3718 3719 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 <p>where <em>V</em> is the volume of the rectangular prism, <span class="formula"><i>ρ</i><sub><span class="mbox">pipe</span></sub></span> 3720 3721 is the scattering length of the 3721 parallelepiped, <span class="formula"><i> Ï</i><sub><span class="mbox">solvent</span></sub></span>3722 parallelepiped, <span class="formula"><i>ρ</i><sub><span class="mbox">solvent</span></sub></span> 3722 3723 is the scattering length of the solvent, and 3723 3724 (if the data are in absolute scale) scale represents the volume fraction (which is unitless) .</p> … … 3742 3743 </tr> 3743 3744 <tr><td>short_side</td> 3744 <td> â«</td>3745 <td>Å</td> 3745 3746 <td>35</td> 3746 3747 </tr> … … 3754 3755 </tr> 3755 3756 <tr><td>sldPipe</td> 3756 <td> â«<sup>-2</sup></td>3757 <td>Å<sup>-2</sup></td> 3757 3758 <td>6.3e-6</td> 3758 3759 </tr> 3759 3760 <tr><td>sldSolv</td> 3760 <td> â«<sup>-2</sup></td>3761 <td>Å<sup>-2</sup></td> 3761 3762 <td>1.0e-6</td> 3762 3763 </tr> … … 3793 3794 <p>The 1D scattering intensity for this model is calculated by forming the difference of the 3794 3795 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 each direction by the same length increment 2 Δ (Nayuk, 2012).</p> 3796 3797 <p>As in the case of the massive parallelepiped, the scattering amplitude is computed for a particular 3797 3798 orientation of the parallelepiped with respect to the scattering vector and then averaged over all … … 3800 3801 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrism_1.png" alt="" /></span></p> 3801 3802 3802 <p>where Ξ is the angle between the <em>z</em> axis and the longest axis of the parallelepiped, Ï3803 <p>where θ is the angle between the <em>z</em> axis and the longest axis of the parallelepiped, φ 3803 3804 is the angle between the scattering vector (lying in the <em>xy</em> plane) and the <em>y</em> axis, and:</p> 3804 3805 … … 3806 3807 3807 3808 <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€â3809 ≤â 3809 3810 <i>B</i>â 3810 â€â3811 ≤â 3811 3812 <i>C</i></span> 3812 3813 , … … 3819 3820 <p style="text-align: center;" align="center"><span style="position: relative; top: 6pt;"><img src="img/RectangularHollowPrism_4.png" alt="" /></span></p> 3820 3821 3821 <p>where <span class="formula"><i> Ï</i><sub><span class="mbox">pipe</span></sub></span>3822 <p>where <span class="formula"><i>ρ</i><sub><span class="mbox">pipe</span></sub></span> 3822 3823 is the scattering length of the 3823 parallelepiped, <span class="formula"><i> Ï</i><sub><span class="mbox">solvent</span></sub></span>3824 parallelepiped, <span class="formula"><i>ρ</i><sub><span class="mbox">solvent</span></sub></span> 3824 3825 is the scattering length of the solvent, and 3825 3826 (if the data are in absolute scale) scale represents the volume fraction (which is unitless) .</p> … … 3845 3846 </tr> 3846 3847 <tr><td>short_side</td> 3847 <td> â«</td>3848 <td>Å</td> 3848 3849 <td>35</td> 3849 3850 </tr> … … 3857 3858 </tr> 3858 3859 <tr><td>thickness</td> 3859 <td> â«</td>3860 <td>Å</td> 3860 3861 <td>1</td> 3861 3862 </tr> 3862 3863 <tr><td>sldPipe</td> 3863 <td> â«<sup>-2</sup></td>3864 <td>Å<sup>-2</sup></td> 3864 3865 <td>6.3e-6</td> 3865 3866 </tr> 3866 3867 <tr><td>sldSolv</td> 3867 <td> â«<sup>-2</sup></td>3868 <td>Å<sup>-2</sup></td> 3868 3869 <td>1.0e-6</td> 3869 3870 </tr>
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