[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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| 6 | **The thickness and the scattering length density of the shell or "rim" |
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| 7 | can be different on all three (pairs) of faces.** |
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| 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 | |
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| 19 | The function calculated is the form factor of the rectangular solid below. |
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[500128b] | 20 | The core of the solid is defined by the dimensions $A$, $B$, $C$ such that |
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| 21 | $A < B < C$. |
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[44bd2be] | 22 | |
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[2f0c07d] | 23 | .. image:: img/core_shell_parallelepiped_geometry.jpg |
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[44bd2be] | 24 | |
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[500128b] | 25 | There are rectangular "slabs" of thickness $t_A$ that add to the $A$ dimension |
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| 26 | (on the $BC$ faces). There are similar slabs on the $AC$ $(=t_B)$ and $AB$ |
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| 27 | $(=t_C)$ faces. The projection in the $AB$ plane is then |
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[44bd2be] | 28 | |
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| 29 | .. image:: img/core_shell_parallelepiped_projection.jpg |
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| 30 | |
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| 31 | The volume of the solid is |
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| 32 | |
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| 33 | .. math:: |
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| 34 | |
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| 35 | V = ABC + 2t_ABC + 2t_BAC + 2t_CAB |
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| 36 | |
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| 37 | **meaning that there are "gaps" at the corners of the solid.** |
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| 38 | |
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[5810f00] | 39 | The intensity calculated follows the :ref:`parallelepiped` model, with the |
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| 40 | core-shell intensity being calculated as the square of the sum of the |
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| 41 | amplitudes of the core and shell, in the same manner as a core-shell model. |
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[44bd2be] | 42 | |
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[5810f00] | 43 | .. math:: |
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| 44 | |
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| 45 | F_{a}(Q,\alpha,\beta)= |
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| 46 | \Bigg(\frac{sin(Q(L_A+2t_A)/2sin\alpha sin\beta)}{Q(L_A+2t_A)/2sin\alpha |
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| 47 | sin\beta)} |
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| 48 | - \frac{sin(QL_A/2sin\alpha sin\beta)}{QL_A/2sin\alpha sin\beta)} \Bigg) |
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| 49 | + \frac{sin(QL_B/2sin\alpha sin\beta)}{QL_B/2sin\alpha sin\beta)} |
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| 50 | + \frac{sin(QL_C/2sin\alpha sin\beta)}{QL_C/2sin\alpha sin\beta)} |
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| 51 | |
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| 52 | .. note:: |
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| 53 | |
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| 54 | For the calculation of the form factor to be valid, the sides of the solid |
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| 55 | MUST be chosen such that** $A < B < C$. |
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| 56 | If this inequality is not satisfied, the model will not report an error, |
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| 57 | but the calculation will not be correct and thus the result wrong. |
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[44bd2be] | 58 | |
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| 59 | FITTING NOTES |
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| 60 | If the scale is set equal to the particle volume fraction, |phi|, the returned |
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[500128b] | 61 | value is the scattered intensity per unit volume, $I(q) = \phi P(q)$. |
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[5810f00] | 62 | However, **no interparticle interference effects are included in this |
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| 63 | calculation.** |
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[44bd2be] | 64 | |
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| 65 | There are many parameters in this model. Hold as many fixed as possible with |
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| 66 | known values, or you will certainly end up at a solution that is unphysical. |
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| 67 | |
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| 68 | Constraints must be applied during fitting to ensure that the inequality |
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[500128b] | 69 | $A < B < C$ is not violated. The calculation will not report an error, |
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[44bd2be] | 70 | but the results will not be correct. |
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| 71 | |
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| 72 | The returned value is in units of |cm^-1|, on absolute scale. |
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| 73 | |
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| 74 | NB: The 2nd virial coefficient of the core_shell_parallelepiped is calculated |
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| 75 | based on the the averaged effective radius $(=\sqrt{(A+2t_A)(B+2t_B)/\pi})$ |
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| 76 | and length $(C+2t_C)$ values, and used as the effective radius |
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[500128b] | 77 | for $S(Q)$ when $P(Q) * S(Q)$ is applied. |
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[44bd2be] | 78 | |
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| 79 | To provide easy access to the orientation of the parallelepiped, we define the |
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[500128b] | 80 | axis of the cylinder using three angles $\theta$, $\phi$ and $\Psi$. |
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[2f0c07d] | 81 | (see :ref:`cylinder orientation <cylinder-angle-definition>`). |
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[500128b] | 82 | The angle $\Psi$ is the rotational angle around the *long_c* axis against the |
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| 83 | $q$ plane. For example, $\Psi = 0$ when the *short_b* axis is parallel to the |
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[44bd2be] | 84 | *x*-axis of the detector. |
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| 85 | |
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[2f0c07d] | 86 | .. figure:: img/parallelepiped_angle_definition.jpg |
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[44bd2be] | 87 | |
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| 88 | Definition of the angles for oriented core-shell parallelepipeds. |
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| 89 | |
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[2f0c07d] | 90 | .. figure:: img/parallelepiped_angle_projection.jpg |
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[44bd2be] | 91 | |
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| 92 | Examples of the angles for oriented core-shell parallelepipeds against the |
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| 93 | detector plane. |
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| 94 | |
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| 95 | Validation |
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| 96 | ---------- |
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| 97 | |
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| 98 | The model uses the form factor calculations implemented in a c-library provided |
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| 99 | by the NIST Center for Neutron Research (Kline, 2006). |
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| 100 | |
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[aa2edb2] | 101 | References |
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| 102 | ---------- |
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[44bd2be] | 103 | |
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[5810f00] | 104 | .. [#] P Mittelbach and G Porod, *Acta Physica Austriaca*, 14 (1961) 185-211 |
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| 105 | Equations (1), (13-14). (in German) |
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| 106 | .. [#] D Singh (2009). *Small angle scattering studies of self assembly in |
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| 107 | lipid mixtures*, John's Hopkins University Thesis (2009) 223-225. `Available |
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| 108 | from Proquest <http://search.proquest.com/docview/304915826?accountid |
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| 109 | =26379>`_ |
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| 110 | |
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| 111 | Authorship and Verification |
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| 112 | ---------------------------- |
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[44bd2be] | 113 | |
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[5810f00] | 114 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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| 115 | * **Last Modified by:** Paul Butler **Date:** September 30, 2016 |
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| 116 | * **Last Reviewed by:** Miguel Gonzales **Date:** March 21, 2016 |
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[44bd2be] | 117 | """ |
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| 118 | |
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| 119 | import numpy as np |
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| 120 | from numpy import pi, inf, sqrt |
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| 121 | |
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| 122 | name = "core_shell_parallelepiped" |
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| 123 | title = "Rectangular solid with a core-shell structure." |
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| 124 | description = """ |
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| 125 | P(q)= |
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| 126 | """ |
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| 127 | category = "shape:parallelepiped" |
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| 128 | |
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| 129 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 130 | parameters = [["sld_core", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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[44bd2be] | 131 | "Parallelepiped core scattering length density"], |
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[42356c8] | 132 | ["sld_a", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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[44bd2be] | 133 | "Parallelepiped A rim scattering length density"], |
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[42356c8] | 134 | ["sld_b", "1e-6/Ang^2", 4, [-inf, inf], "sld", |
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[44bd2be] | 135 | "Parallelepiped B rim scattering length density"], |
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[42356c8] | 136 | ["sld_c", "1e-6/Ang^2", 2, [-inf, inf], "sld", |
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[44bd2be] | 137 | "Parallelepiped C rim scattering length density"], |
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[42356c8] | 138 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", |
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[44bd2be] | 139 | "Solvent scattering length density"], |
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[2222134] | 140 | ["length_a", "Ang", 35, [0, inf], "volume", |
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[44bd2be] | 141 | "Shorter side of the parallelepiped"], |
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[2222134] | 142 | ["length_b", "Ang", 75, [0, inf], "volume", |
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[44bd2be] | 143 | "Second side of the parallelepiped"], |
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[2222134] | 144 | ["length_c", "Ang", 400, [0, inf], "volume", |
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[44bd2be] | 145 | "Larger side of the parallelepiped"], |
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[2222134] | 146 | ["thick_rim_a", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 147 | "Thickness of A rim"], |
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[2222134] | 148 | ["thick_rim_b", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 149 | "Thickness of B rim"], |
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[2222134] | 150 | ["thick_rim_c", "Ang", 10, [0, inf], "volume", |
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[44bd2be] | 151 | "Thickness of C rim"], |
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| 152 | ["theta", "degrees", 0, [-inf, inf], "orientation", |
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| 153 | "In plane angle"], |
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| 154 | ["phi", "degrees", 0, [-inf, inf], "orientation", |
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| 155 | "Out of plane angle"], |
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| 156 | ["psi", "degrees", 0, [-inf, inf], "orientation", |
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| 157 | "Rotation angle around its own c axis against q plane"], |
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| 158 | ] |
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| 159 | |
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[43b7eea] | 160 | source = ["lib/gauss76.c", "core_shell_parallelepiped.c"] |
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[44bd2be] | 161 | |
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| 162 | |
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[2222134] | 163 | def ER(length_a, length_b, length_c, thick_rim_a, thick_rim_b, thick_rim_c): |
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[44bd2be] | 164 | """ |
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| 165 | Return equivalent radius (ER) |
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| 166 | """ |
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| 167 | |
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| 168 | # surface average radius (rough approximation) |
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[2222134] | 169 | surf_rad = sqrt((length_a + 2.0*thick_rim_a) * (length_b + 2.0*thick_rim_b) / pi) |
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[44bd2be] | 170 | |
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[2222134] | 171 | height = length_c + 2.0*thick_rim_c |
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[44bd2be] | 172 | |
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| 173 | ddd = 0.75 * surf_rad * (2 * surf_rad * height + (height + surf_rad) * (height + pi * surf_rad)) |
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| 174 | return 0.5 * (ddd) ** (1. / 3.) |
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| 175 | |
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| 176 | # VR defaults to 1.0 |
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| 177 | |
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| 178 | # parameters for demo |
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| 179 | demo = dict(scale=1, background=0.0, |
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[14838a3] | 180 | sld_core=1, sld_a=2, sld_b=4, sld_c=2, sld_solvent=6, |
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[2222134] | 181 | length_a=35, length_b=75, length_c=400, |
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| 182 | thick_rim_a=10, thick_rim_b=10, thick_rim_c=10, |
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[44bd2be] | 183 | theta=0, phi=0, psi=0, |
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[2222134] | 184 | length_a_pd=0.1, length_a_pd_n=1, |
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| 185 | length_b_pd=0.1, length_b_pd_n=1, |
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| 186 | length_c_pd=0.1, length_c_pd_n=1, |
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| 187 | thick_rim_a_pd=0.1, thick_rim_a_pd_n=1, |
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| 188 | thick_rim_b_pd=0.1, thick_rim_b_pd_n=1, |
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| 189 | thick_rim_c_pd=0.1, thick_rim_c_pd_n=1, |
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[44bd2be] | 190 | theta_pd=10, theta_pd_n=1, |
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| 191 | phi_pd=10, phi_pd_n=1, |
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[14838a3] | 192 | psi_pd=10, psi_pd_n=1) |
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[44bd2be] | 193 | |
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| 194 | qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5) |
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[6dd90c1] | 195 | tests = [[{}, 0.2, 0.533149288477], |
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| 196 | [{}, [0.2], [0.533149288477]], |
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| 197 | [{'theta':10.0, 'phi':10.0}, (qx, qy), 0.032102135569], |
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| 198 | [{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.032102135569]], |
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[44bd2be] | 199 | ] |
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| 200 | del qx, qy # not necessary to delete, but cleaner |
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