[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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[cb0dc22] | 7 | "rim" can be different on each (pair) of faces. However at this time |
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[1f65db5] | 8 | the 1D calculation does **NOT** actually calculate a c face rim despite the presence of |
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| 9 | the parameter. Some other aspects of the 1D calculation may be wrong. |
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[cb0dc22] | 10 | |
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| 11 | .. note:: |
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[1916c52] | 12 | This model was originally ported from NIST IGOR macros. However, it is not |
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| 13 | yet fully understood by the SasView developers and is currently under review. |
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[44bd2be] | 14 | |
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[500128b] | 15 | The form factor is normalized by the particle volume $V$ such that |
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[44bd2be] | 16 | |
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[500128b] | 17 | .. math:: |
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| 18 | |
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| 19 | I(q) = \text{scale}\frac{\langle f^2 \rangle}{V} + \text{background} |
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[44bd2be] | 20 | |
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[500128b] | 21 | where $\langle \ldots \rangle$ is an average over all possible orientations |
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| 22 | of the rectangular solid. |
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[44bd2be] | 23 | |
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| 24 | |
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| 25 | The function calculated is the form factor of the rectangular solid below. |
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[500128b] | 26 | The core of the solid is defined by the dimensions $A$, $B$, $C$ such that |
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| 27 | $A < B < C$. |
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[44bd2be] | 28 | |
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[2f0c07d] | 29 | .. image:: img/core_shell_parallelepiped_geometry.jpg |
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[44bd2be] | 30 | |
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[500128b] | 31 | There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension |
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| 32 | (on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$ |
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| 33 | $(=t_C)$ faces. The projection in the $AB$ plane is then |
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[44bd2be] | 34 | |
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[1916c52] | 35 | .. image:: img/core_shell_parallelepiped_projection.jpg |
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[44bd2be] | 36 | |
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| 37 | The volume of the solid is |
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| 38 | |
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| 39 | .. math:: |
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| 40 | |
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| 41 | V = ABC + 2t_ABC + 2t_BAC + 2t_CAB |
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| 42 | |
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[cb0dc22] | 43 | **meaning that there are "gaps" at the corners of the solid.** Again note that |
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[8f04da4] | 44 | $t_C = 0$ currently. |
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[44bd2be] | 45 | |
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[5810f00] | 46 | The intensity calculated follows the :ref:`parallelepiped` model, with the |
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| 47 | core-shell intensity being calculated as the square of the sum of the |
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| 48 | amplitudes of the core and shell, in the same manner as a core-shell model. |
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[44bd2be] | 49 | |
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[5810f00] | 50 | .. math:: |
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| 51 | |
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| 52 | F_{a}(Q,\alpha,\beta)= |
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[1916c52] | 53 | \left[\frac{\sin(\tfrac{1}{2}Q(L_A+2t_A)\sin\alpha \sin\beta)}{\tfrac{1}{2}Q(L_A+2t_A)\sin\alpha\sin\beta} |
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[14207bb] | 54 | - \frac{\sin(\tfrac{1}{2}QL_A\sin\alpha \sin\beta)}{\tfrac{1}{2}QL_A\sin\alpha \sin\beta} \right] |
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| 55 | \left[\frac{\sin(\tfrac{1}{2}QL_B\sin\alpha \sin\beta)}{\tfrac{1}{2}QL_B\sin\alpha \sin\beta} \right] |
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| 56 | \left[\frac{\sin(\tfrac{1}{2}QL_C\sin\alpha \sin\beta)}{\tfrac{1}{2}QL_C\sin\alpha \sin\beta} \right] |
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[5810f00] | 57 | |
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| 58 | .. note:: |
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| 59 | |
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[1916c52] | 60 | Why does t_B not appear in the above equation? |
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[5810f00] | 61 | For the calculation of the form factor to be valid, the sides of the solid |
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[1916c52] | 62 | MUST (perhaps not any more?) be chosen such that** $A < B < C$. |
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[5810f00] | 63 | If this inequality is not satisfied, the model will not report an error, |
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| 64 | but the calculation will not be correct and thus the result wrong. |
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[44bd2be] | 65 | |
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| 66 | FITTING NOTES |
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[92dfe0c] | 67 | If the scale is set equal to the particle volume fraction, $\phi$, the returned |
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[500128b] | 68 | value is the scattered intensity per unit volume, $I(q) = \phi P(q)$. |
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[5810f00] | 69 | However, **no interparticle interference effects are included in this |
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| 70 | calculation.** |
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[44bd2be] | 71 | |
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| 72 | There are many parameters in this model. Hold as many fixed as possible with |
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| 73 | known values, or you will certainly end up at a solution that is unphysical. |
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| 74 | |
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| 75 | Constraints must be applied during fitting to ensure that the inequality |
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[500128b] | 76 | $A < B < C$ is not violated. The calculation will not report an error, |
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[44bd2be] | 77 | but the results will not be correct. |
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| 78 | |
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| 79 | The returned value is in units of |cm^-1|, on absolute scale. |
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| 80 | |
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| 81 | NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated |
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| 82 | based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ |
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[15a90c1] | 83 | and length $(C+2t_C)$ values, after appropriately |
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[8f04da4] | 84 | sorting the three dimensions to give an oblate or prolate particle, to give an |
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[15a90c1] | 85 | effective radius, for $S(Q)$ when $P(Q) * S(Q)$ is applied. |
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[44bd2be] | 86 | |
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| 87 | To provide easy access to the orientation of the parallelepiped, we define the |
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[500128b] | 88 | axis of the cylinder using three angles $\theta$, $\phi$ and $\Psi$. |
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[2f0c07d] | 89 | (see :ref:`cylinder orientation <cylinder-angle-definition>`). |
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[500128b] | 90 | The angle $\Psi$ is the rotational angle around the *long_c* axis against the |
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| 91 | $q$ plane. For example, $\Psi = 0$ when the *short_b* axis is parallel to the |
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[44bd2be] | 92 | *x*-axis of the detector. |
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| 93 | |
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[15a90c1] | 94 | .. figure:: img/parallelepiped_angle_definition.png |
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[44bd2be] | 95 | |
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| 96 | Definition of the angles for oriented core-shell parallelepipeds. |
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| 97 | |
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[1916c52] | 98 | .. figure:: img/parallelepiped_angle_projection.png |
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[44bd2be] | 99 | |
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| 100 | Examples of the angles for oriented core-shell parallelepipeds against the |
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| 101 | detector plane. |
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| 102 | |
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[aa2edb2] | 103 | References |
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| 104 | ---------- |
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[44bd2be] | 105 | |
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[5810f00] | 106 | .. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
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| 107 | Equations (1), (13-14). (in German) |
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| 108 | .. [#] D Singh (2009). *Small angle scattering studies of self assembly in |
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| 109 | lipid mixtures*, John's Hopkins University Thesis (2009) 223-225. `Available |
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| 110 | from Proquest <http://search.proquest.com/docview/304915826?accountid |
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| 111 | =26379>`_ |
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| 112 | |
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| 113 | Authorship and Verification |
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| 114 | ---------------------------- |
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[44bd2be] | 115 | |
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[5810f00] | 116 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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[cb0dc22] | 117 | * **Converted to sasmodels by:** Miguel Gonzales **Date:** February 26, 2016 |
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| 118 | * **Last Modified by:** Wojciech Potrzebowski **Date:** January 11, 2017 |
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| 119 | * **Currently Under review by:** Paul Butler |
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[44bd2be] | 120 | """ |
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| 121 | |
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| 122 | import numpy as np |
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[14207bb] | 123 | from numpy import pi, inf, sqrt, cos, sin |
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[44bd2be] | 124 | |
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| 125 | name = "core_shell_parallelepiped" |
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| 126 | title = "Rectangular solid with a core-shell structure." |
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| 127 | description = """ |
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[8f04da4] | 128 | P(q)= |
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[44bd2be] | 129 | """ |
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| 130 | category = "shape:parallelepiped" |
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| 131 | |
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| 132 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 133 | parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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[44bd2be] | 134 | "Parallelepiped core scattering length density"], |
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[42356c8] | 135 | ["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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[44bd2be] | 136 | "Parallelepiped A rim scattering length density"], |
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[42356c8] | 137 | ["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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[44bd2be] | 138 | "Parallelepiped B rim scattering length density"], |
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[42356c8] | 139 | ["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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[44bd2be] | 140 | "Parallelepiped C rim scattering length density"], |
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[42356c8] | 141 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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[44bd2be] | 142 | "Solvent scattering length density"], |
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[2222134] | 143 | ["length_a", "Ang", 35, [0, inf], "volume", |
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[44bd2be] | 144 | "Shorter side of the parallelepiped"], |
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[2222134] | 145 | ["length_b", "Ang", 75, [0, inf], "volume", |
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[44bd2be] | 146 | "Second side of the parallelepiped"], |
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[2222134] | 147 | ["length_c", "Ang", 400, [0, inf], "volume", |
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[44bd2be] | 148 | "Larger side of the parallelepiped"], |
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[2222134] | 149 | ["thick_rim_a", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 150 | "Thickness of A rim"], |
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[2222134] | 151 | ["thick_rim_b", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 152 | "Thickness of B rim"], |
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[2222134] | 153 | ["thick_rim_c", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 154 | "Thickness of C rim"], |
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[9b79f29] | 155 | ["theta", "degrees", 0, [-360, 360], "orientation", |
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| 156 | "c axis to beam angle"], |
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| 157 | ["phi", "degrees", 0, [-360, 360], "orientation", |
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| 158 | "rotation about beam"], |
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| 159 | ["psi", "degrees", 0, [-360, 360], "orientation", |
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| 160 | "rotation about c axis"], |
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[44bd2be] | 161 | ] |
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| 162 | |
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[43b7eea] | 163 | source = ["lib/gauss76.c", "core_shell_parallelepiped.c"] |
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[44bd2be] | 164 | |
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| 165 | |
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[2222134] | 166 | def ER(length_a, length_b, length_c, thick_rim_a, thick_rim_b, thick_rim_c): |
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[44bd2be] | 167 | """ |
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| 168 | Return equivalent radius (ER) |
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| 169 | """ |
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| 170 | |
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| 171 | # surface average radius (rough approximation) |
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[2222134] | 172 | surf_rad = sqrt((length_a + 2.0*thick_rim_a) * (length_b + 2.0*thick_rim_b) / pi) |
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[44bd2be] | 173 | |
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[2222134] | 174 | height = length_c + 2.0*thick_rim_c |
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[44bd2be] | 175 | |
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| 176 | ddd = 0.75 * surf_rad * (2 * surf_rad * height + (height + surf_rad) * (height + pi * surf_rad)) |
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| 177 | return 0.5 * (ddd) ** (1. / 3.) |
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| 178 | |
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| 179 | # VR defaults to 1.0 |
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| 180 | |
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[8f04da4] | 181 | def random(): |
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| 182 | import numpy as np |
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| 183 | outer = 10**np.random.uniform(1, 4.7, size=3) |
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| 184 | thick = np.random.beta(0.5, 0.5, size=3)*(outer-2) + 1 |
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| 185 | length = outer - thick |
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| 186 | pars = dict( |
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| 187 | length_a=length[0], |
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| 188 | length_b=length[1], |
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| 189 | length_c=length[2], |
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| 190 | thick_rim_a=thick[0], |
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| 191 | thick_rim_b=thick[1], |
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| 192 | thick_rim_c=thick[2], |
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| 193 | ) |
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| 194 | return pars |
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| 195 | |
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[44bd2be] | 196 | # parameters for demo |
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| 197 | demo = dict(scale=1, background=0.0, |
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[14838a3] | 198 | sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6, |
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[2222134] | 199 | length_a=35, length_b=75, length_c=400, |
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| 200 | thick_rim_a=10, thick_rim_b=10, thick_rim_c=10, |
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[44bd2be] | 201 | theta=0, phi=0, psi=0, |
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[2222134] | 202 | length_a_pd=0.1, length_a_pd_n=1, |
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| 203 | length_b_pd=0.1, length_b_pd_n=1, |
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| 204 | length_c_pd=0.1, length_c_pd_n=1, |
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| 205 | thick_rim_a_pd=0.1, thick_rim_a_pd_n=1, |
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| 206 | thick_rim_b_pd=0.1, thick_rim_b_pd_n=1, |
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| 207 | thick_rim_c_pd=0.1, thick_rim_c_pd_n=1, |
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[44bd2be] | 208 | theta_pd=10, theta_pd_n=1, |
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| 209 | phi_pd=10, phi_pd_n=1, |
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[14838a3] | 210 | psi_pd=10, psi_pd_n=1) |
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[44bd2be] | 211 | |
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[14207bb] | 212 | # rkh 7/4/17 add random unit test for 2d, note make all params different, 2d values not tested against other codes or models |
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| 213 | qx, qy = 0.2 * cos(pi/6.), 0.2 * sin(pi/6.) |
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[6dd90c1] | 214 | tests = [[{}, 0.2, 0.533149288477], |
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| 215 | [{}, [0.2], [0.533149288477]], |
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[14207bb] | 216 | [{'theta':10.0, 'phi':20.0}, (qx, qy), 0.0853299803222], |
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| 217 | [{'theta':10.0, 'phi':20.0}, [(qx, qy)], [0.0853299803222]], |
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[44bd2be] | 218 | ] |
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| 219 | del qx, qy # not necessary to delete, but cleaner |
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