[44bd2be] | 1 | r""" |
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[5810f00] | 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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[44bd2be] | 5 | Calculates the form factor for a rectangular solid with a core-shell structure. |
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[8f04da4] | 6 | The thickness and the scattering length density of the shell or |
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[393facf] | 7 | "rim" can be different on each (pair) of faces. |
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[cb0dc22] | 8 | |
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[500128b] | 9 | The form factor is normalized by the particle volume $V$ such that |
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[44bd2be] | 10 | |
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[500128b] | 11 | .. math:: |
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| 12 | |
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| 13 | I(q) = \text{scale}\frac{\langle f^2 \rangle}{V} + \text{background} |
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[44bd2be] | 14 | |
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[500128b] | 15 | where $\langle \ldots \rangle$ is an average over all possible orientations |
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| 16 | of the rectangular solid. |
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[44bd2be] | 17 | |
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| 18 | The function calculated is the form factor of the rectangular solid below. |
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[500128b] | 19 | The core of the solid is defined by the dimensions $A$, $B$, $C$ such that |
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| 20 | $A < B < C$. |
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[44bd2be] | 21 | |
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[2f0c07d] | 22 | .. image:: img/core_shell_parallelepiped_geometry.jpg |
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[44bd2be] | 23 | |
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[500128b] | 24 | There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension |
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| 25 | (on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$ |
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| 26 | $(=t_C)$ faces. The projection in the $AB$ plane is then |
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[44bd2be] | 27 | |
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[1916c52] | 28 | .. image:: img/core_shell_parallelepiped_projection.jpg |
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[44bd2be] | 29 | |
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| 30 | The volume of the solid is |
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| 31 | |
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| 32 | .. math:: |
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| 33 | |
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| 34 | V = ABC + 2t_ABC + 2t_BAC + 2t_CAB |
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| 35 | |
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[393facf] | 36 | **meaning that there are "gaps" at the corners of the solid.** |
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[44bd2be] | 37 | |
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[5810f00] | 38 | The intensity calculated follows the :ref:`parallelepiped` model, with the |
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| 39 | core-shell intensity being calculated as the square of the sum of the |
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[393facf] | 40 | amplitudes of the core and the slabs on the edges. |
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| 41 | |
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[4493288] | 42 | the scattering amplitude is computed for a particular orientation of the |
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| 43 | core-shell parallelepiped with respect to the scattering vector and then |
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| 44 | averaged over all possible orientations, where $\alpha$ is the angle between |
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| 45 | the $z$ axis and the $C$ axis of the parallelepiped, $\beta$ is |
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| 46 | the angle between projection of the particle in the $xy$ detector plane |
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| 47 | and the $y$ axis. |
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[44bd2be] | 48 | |
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[5810f00] | 49 | .. math:: |
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[4493288] | 50 | |
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| 51 | F(Q) |
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| 52 | &= (\rho_\text{core}-\rho_\text{solvent}) |
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| 53 | S(Q_A, A) S(Q_B, B) S(Q_C, C) \\ |
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| 54 | &+ (\rho_\text{A}-\rho_\text{solvent}) |
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| 55 | \left[S(Q_A, A+2t_A) - S(Q_A, Q)\right] S(Q_B, B) S(Q_C, C) \\ |
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| 56 | &+ (\rho_\text{B}-\rho_\text{solvent}) |
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| 57 | S(Q_A, A) \left[S(Q_B, B+2t_B) - S(Q_B, B)\right] S(Q_C, C) \\ |
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| 58 | &+ (\rho_\text{C}-\rho_\text{solvent}) |
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| 59 | S(Q_A, A) S(Q_B, B) \left[S(Q_C, C+2t_C) - S(Q_C, C)\right] |
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[393facf] | 60 | |
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| 61 | with |
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[5810f00] | 62 | |
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[393facf] | 63 | .. math:: |
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[5810f00] | 64 | |
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[4493288] | 65 | S(Q, L) = L \frac{\sin \tfrac{1}{2} Q L}{\tfrac{1}{2} Q L} |
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| 66 | |
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| 67 | and |
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| 68 | |
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| 69 | .. math:: |
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[5810f00] | 70 | |
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[4493288] | 71 | Q_A &= \sin\alpha \sin\beta \\ |
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| 72 | Q_B &= \sin\alpha \cos\beta \\ |
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| 73 | Q_C &= \cos\alpha |
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| 74 | |
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| 75 | |
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| 76 | where $\rho_\text{core}$, $\rho_\text{A}$, $\rho_\text{B}$ and $\rho_\text{C}$ |
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| 77 | are the scattering length of the parallelepiped core, and the rectangular |
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| 78 | slabs of thickness $t_A$, $t_B$ and $t_C$, respectively. $\rho_\text{solvent}$ |
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| 79 | is the scattering length of the solvent. |
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[44bd2be] | 80 | |
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| 81 | FITTING NOTES |
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[4493288] | 82 | ~~~~~~~~~~~~~ |
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| 83 | |
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[92dfe0c] | 84 | If the scale is set equal to the particle volume fraction, $\phi$, the returned |
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[4493288] | 85 | value is the scattered intensity per unit volume, $I(q) = \phi P(q)$. However, |
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| 86 | **no interparticle interference effects are included in this calculation.** |
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[44bd2be] | 87 | |
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| 88 | There are many parameters in this model. Hold as many fixed as possible with |
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| 89 | known values, or you will certainly end up at a solution that is unphysical. |
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| 90 | |
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| 91 | The returned value is in units of |cm^-1|, on absolute scale. |
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| 92 | |
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| 93 | NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated |
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| 94 | based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ |
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[4493288] | 95 | and length $(C+2t_C)$ values, after appropriately sorting the three dimensions |
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| 96 | to give an oblate or prolate particle, to give an effective radius, |
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| 97 | for $S(Q)$ when $P(Q) * S(Q)$ is applied. |
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[44bd2be] | 98 | |
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[904cd9c] | 99 | For 2d data the orientation of the particle is required, described using |
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[4493288] | 100 | angles $\theta$, $\phi$ and $\Psi$ as in the diagrams below, for further |
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| 101 | details of the calculation and angular dispersions see :ref:`orientation`. |
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[904cd9c] | 102 | The angle $\Psi$ is the rotational angle around the *long_c* axis. For example, |
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[eda8b30] | 103 | $\Psi = 0$ when the *short_b* axis is parallel to the *x*-axis of the detector. |
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[44bd2be] | 104 | |
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[4493288] | 105 | For 2d, constraints must be applied during fitting to ensure that the |
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| 106 | inequality $A < B < C$ is not violated, and hence the correct definition |
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| 107 | of angles is preserved. The calculation will not report an error, |
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[393facf] | 108 | but the results may be not correct. |
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| 109 | |
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[15a90c1] | 110 | .. figure:: img/parallelepiped_angle_definition.png |
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[44bd2be] | 111 | |
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| 112 | Definition of the angles for oriented core-shell parallelepipeds. |
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[2d81cfe] | 113 | Note that rotation $\theta$, initially in the $xz$ plane, is carried |
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| 114 | out first, then rotation $\phi$ about the $z$ axis, finally rotation |
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| 115 | $\Psi$ is now around the axis of the cylinder. The neutron or X-ray |
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| 116 | beam is along the $z$ axis. |
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[44bd2be] | 117 | |
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[1916c52] | 118 | .. figure:: img/parallelepiped_angle_projection.png |
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[44bd2be] | 119 | |
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| 120 | Examples of the angles for oriented core-shell parallelepipeds against the |
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| 121 | detector plane. |
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| 122 | |
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[aa2edb2] | 123 | References |
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| 124 | ---------- |
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[44bd2be] | 125 | |
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[5810f00] | 126 | .. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
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| 127 | Equations (1), (13-14). (in German) |
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| 128 | .. [#] D Singh (2009). *Small angle scattering studies of self assembly in |
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[fc0b7aa] | 129 | lipid mixtures*, Johns Hopkins University Thesis (2009) 223-225. `Available |
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[5810f00] | 130 | from Proquest <http://search.proquest.com/docview/304915826?accountid |
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| 131 | =26379>`_ |
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| 132 | |
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| 133 | Authorship and Verification |
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| 134 | ---------------------------- |
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[44bd2be] | 135 | |
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[5810f00] | 136 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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[cb0dc22] | 137 | * **Converted to sasmodels by:** Miguel Gonzales **Date:** February 26, 2016 |
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| 138 | * **Last Modified by:** Wojciech Potrzebowski **Date:** January 11, 2017 |
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| 139 | * **Currently Under review by:** Paul Butler |
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[44bd2be] | 140 | """ |
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| 141 | |
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| 142 | import numpy as np |
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[14207bb] | 143 | from numpy import pi, inf, sqrt, cos, sin |
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[44bd2be] | 144 | |
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| 145 | name = "core_shell_parallelepiped" |
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| 146 | title = "Rectangular solid with a core-shell structure." |
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| 147 | description = """ |
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[8f04da4] | 148 | P(q)= |
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[44bd2be] | 149 | """ |
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| 150 | category = "shape:parallelepiped" |
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| 151 | |
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| 152 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 153 | parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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[44bd2be] | 154 | "Parallelepiped core scattering length density"], |
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[42356c8] | 155 | ["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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[44bd2be] | 156 | "Parallelepiped A rim scattering length density"], |
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[42356c8] | 157 | ["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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[44bd2be] | 158 | "Parallelepiped B rim scattering length density"], |
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[42356c8] | 159 | ["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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[44bd2be] | 160 | "Parallelepiped C rim scattering length density"], |
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[42356c8] | 161 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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[44bd2be] | 162 | "Solvent scattering length density"], |
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[2222134] | 163 | ["length_a", "Ang", 35, [0, inf], "volume", |
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[44bd2be] | 164 | "Shorter side of the parallelepiped"], |
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[2222134] | 165 | ["length_b", "Ang", 75, [0, inf], "volume", |
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[44bd2be] | 166 | "Second side of the parallelepiped"], |
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[2222134] | 167 | ["length_c", "Ang", 400, [0, inf], "volume", |
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[44bd2be] | 168 | "Larger side of the parallelepiped"], |
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[2222134] | 169 | ["thick_rim_a", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 170 | "Thickness of A rim"], |
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[2222134] | 171 | ["thick_rim_b", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 172 | "Thickness of B rim"], |
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[2222134] | 173 | ["thick_rim_c", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 174 | "Thickness of C rim"], |
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[9b79f29] | 175 | ["theta", "degrees", 0, [-360, 360], "orientation", |
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| 176 | "c axis to beam angle"], |
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| 177 | ["phi", "degrees", 0, [-360, 360], "orientation", |
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| 178 | "rotation about beam"], |
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| 179 | ["psi", "degrees", 0, [-360, 360], "orientation", |
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| 180 | "rotation about c axis"], |
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[44bd2be] | 181 | ] |
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| 182 | |
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[43b7eea] | 183 | source = ["lib/gauss76.c", "core_shell_parallelepiped.c"] |
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[44bd2be] | 184 | |
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| 185 | |
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[2222134] | 186 | def ER(length_a, length_b, length_c, thick_rim_a, thick_rim_b, thick_rim_c): |
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[44bd2be] | 187 | """ |
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| 188 | Return equivalent radius (ER) |
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| 189 | """ |
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| 190 | |
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| 191 | # surface average radius (rough approximation) |
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[4493288] | 192 | surf_rad = sqrt((length_a + 2.0*thick_rim_a) |
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| 193 | * (length_b + 2.0*thick_rim_b) / pi) |
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[44bd2be] | 194 | |
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[2222134] | 195 | height = length_c + 2.0*thick_rim_c |
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[44bd2be] | 196 | |
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[4493288] | 197 | ddd = (0.75 * surf_rad * (2 * surf_rad * height |
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| 198 | + (height + surf_rad) * (height + pi * surf_rad))) |
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[44bd2be] | 199 | return 0.5 * (ddd) ** (1. / 3.) |
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| 200 | |
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| 201 | # VR defaults to 1.0 |
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| 202 | |
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[8f04da4] | 203 | def random(): |
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| 204 | outer = 10**np.random.uniform(1, 4.7, size=3) |
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| 205 | thick = np.random.beta(0.5, 0.5, size=3)*(outer-2) + 1 |
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| 206 | length = outer - thick |
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| 207 | pars = dict( |
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| 208 | length_a=length[0], |
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| 209 | length_b=length[1], |
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| 210 | length_c=length[2], |
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| 211 | thick_rim_a=thick[0], |
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| 212 | thick_rim_b=thick[1], |
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| 213 | thick_rim_c=thick[2], |
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| 214 | ) |
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| 215 | return pars |
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| 216 | |
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[44bd2be] | 217 | # parameters for demo |
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| 218 | demo = dict(scale=1, background=0.0, |
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[14838a3] | 219 | sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6, |
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[2222134] | 220 | length_a=35, length_b=75, length_c=400, |
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| 221 | thick_rim_a=10, thick_rim_b=10, thick_rim_c=10, |
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[44bd2be] | 222 | theta=0, phi=0, psi=0, |
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[2222134] | 223 | length_a_pd=0.1, length_a_pd_n=1, |
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| 224 | length_b_pd=0.1, length_b_pd_n=1, |
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| 225 | length_c_pd=0.1, length_c_pd_n=1, |
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| 226 | thick_rim_a_pd=0.1, thick_rim_a_pd_n=1, |
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| 227 | thick_rim_b_pd=0.1, thick_rim_b_pd_n=1, |
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| 228 | thick_rim_c_pd=0.1, thick_rim_c_pd_n=1, |
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[44bd2be] | 229 | theta_pd=10, theta_pd_n=1, |
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| 230 | phi_pd=10, phi_pd_n=1, |
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[14838a3] | 231 | psi_pd=10, psi_pd_n=1) |
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[44bd2be] | 232 | |
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[4493288] | 233 | # rkh 7/4/17 add random unit test for 2d, note make all params different, |
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| 234 | # 2d values not tested against other codes or models |
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[fa70e04] | 235 | if 0: # pak: model rewrite; need to update tests |
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| 236 | qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.) |
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| 237 | tests = [[{}, 0.2, 0.533149288477], |
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[2d81cfe] | 238 | [{}, [0.2], [0.533149288477]], |
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| 239 | [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0853299803222], |
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| 240 | [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0853299803222]], |
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[fa70e04] | 241 | ] |
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| 242 | del qx, qy # not necessary to delete, but cleaner |
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