[deb7ee0] | 1 | # rectangular_prism model |
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| 2 | # Note: model title and parameter table are inserted automatically |
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| 3 | r""" |
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| 4 | Definition |
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| 5 | ---------- |
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| 6 | |
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[6e7d7b6] | 7 | |
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| 8 | This model provides the form factor, $P(q)$, for a hollow rectangular |
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| 9 | prism with infinitely thin walls. It computes only the 1D scattering, not the 2D. |
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[deb7ee0] | 10 | The 1D scattering intensity for this model is calculated according to the |
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[6e7d7b6] | 11 | equations given by Nayuk and Huber\ [#Nayuk2012]_. |
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[deb7ee0] | 12 | |
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| 13 | Assuming a hollow parallelepiped with infinitely thin walls, edge lengths |
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[ab2aea8] | 14 | $A \le B \le C$ and presenting an orientation with respect to the |
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| 15 | scattering vector given by $\theta$ and $\phi$, where $\theta$ is the angle |
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| 16 | between the $z$ axis and the longest axis of the parallelepiped $C$, and |
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| 17 | $\phi$ is the angle between the scattering vector (lying in the $xy$ plane) |
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| 18 | and the $y$ axis, the form factor is given by |
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[deb7ee0] | 19 | |
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| 20 | .. math:: |
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| 21 | |
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[ab2aea8] | 22 | P(q) = \frac{1}{V^2} \frac{2}{\pi} \int_0^{\frac{\pi}{2}} |
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| 23 | \int_0^{\frac{\pi}{2}} [A_L(q)+A_T(q)]^2 \sin\theta\,d\theta\,d\phi |
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[deb7ee0] | 24 | |
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[ab2aea8] | 25 | where |
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[deb7ee0] | 26 | |
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| 27 | .. math:: |
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| 28 | |
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[ab2aea8] | 29 | V &= 2AB + 2AC + 2BC \\ |
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| 30 | A_L(q) &= 8 \times \frac{ |
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| 31 | \sin \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right) |
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| 32 | \sin \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) |
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| 33 | \cos \left( \tfrac{1}{2} q C \cos\theta \right) |
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| 34 | }{q^2 \, \sin^2\theta \, \sin\phi \cos\phi} \\ |
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| 35 | A_T(q) &= A_F(q) \times |
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| 36 | \frac{2\,\sin \left( \tfrac{1}{2} q C \cos\theta \right)}{q\,\cos\theta} |
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[deb7ee0] | 37 | |
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| 38 | and |
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| 39 | |
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| 40 | .. math:: |
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[ab2aea8] | 41 | |
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| 42 | A_F(q) = 4 \frac{ \cos \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right) |
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| 43 | \sin \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) } |
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[deb7ee0] | 44 | {q \, \cos\phi \, \sin\theta} + |
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[ab2aea8] | 45 | 4 \frac{ \sin \left( \tfrac{1}{2} q A \sin\phi \sin\theta \right) |
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| 46 | \cos \left( \tfrac{1}{2} q B \cos\phi \sin\theta \right) } |
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[deb7ee0] | 47 | {q \, \sin\phi \, \sin\theta} |
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| 48 | |
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| 49 | The 1D scattering intensity is then calculated as |
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| 50 | |
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| 51 | .. math:: |
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| 52 | |
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[ab2aea8] | 53 | I(q) = \text{scale} \times V \times (\rho_\text{p} - \rho_\text{solvent})^2 \times P(q) |
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| 54 | |
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[6e7d7b6] | 55 | where $V$ is the surface area of the rectangular prism, $\rho_\text{p}$ |
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| 56 | is the scattering length density of the parallelepiped, $\rho_\text{solvent}$ |
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| 57 | is the scattering length density of the solvent, and (if the data are in |
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| 58 | absolute units) *scale* is related to the total surface area. |
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[deb7ee0] | 59 | |
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| 60 | **The 2D scattering intensity is not computed by this model.** |
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| 61 | |
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| 62 | |
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| 63 | Validation |
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| 64 | ---------- |
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| 65 | |
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| 66 | Validation of the code was conducted by qualitatively comparing the output |
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[6e7d7b6] | 67 | of the 1D model to the curves shown in (Nayuk, 2012\ [#Nayuk2012]_). |
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[deb7ee0] | 68 | |
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[aa2edb2] | 69 | |
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| 70 | References |
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| 71 | ---------- |
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[deb7ee0] | 72 | |
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[6e7d7b6] | 73 | .. [#Nayuk2012] R Nayuk and K Huber, *Z. Phys. Chem.*, 226 (2012) 837-854 |
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[99658f6] | 74 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
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[6e7d7b6] | 75 | |
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| 76 | |
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| 77 | Authorship and Verification |
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| 78 | ---------------------------- |
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| 79 | |
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| 80 | * **Author:** Miguel Gonzales **Date:** February 26, 2016 |
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| 81 | * **Last Modified by:** Paul Kienzle **Date:** October 15, 2016 |
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| 82 | * **Last Reviewed by:** Paul Butler **Date:** September 07, 2018 |
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[deb7ee0] | 83 | """ |
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| 84 | |
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[2d81cfe] | 85 | import numpy as np |
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[deb7ee0] | 86 | from numpy import pi, inf, sqrt |
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| 87 | |
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[3d8283b] | 88 | name = "hollow_rectangular_prism_thin_walls" |
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| 89 | title = "Hollow rectangular parallelepiped with thin walls." |
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[deb7ee0] | 90 | description = """ |
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[3d8283b] | 91 | I(q)= scale*V*(sld - sld_solvent)^2*P(q)+background |
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[deb7ee0] | 92 | with P(q) being the form factor corresponding to a hollow rectangular |
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| 93 | parallelepiped with infinitely thin walls. |
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| 94 | """ |
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| 95 | category = "shape:parallelepiped" |
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| 96 | |
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| 97 | # ["name", "units", default, [lower, upper], "type","description"], |
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[42356c8] | 98 | parameters = [["sld", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", |
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[deb7ee0] | 99 | "Parallelepiped scattering length density"], |
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[42356c8] | 100 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", |
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[deb7ee0] | 101 | "Solvent scattering length density"], |
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[a807206] | 102 | ["length_a", "Ang", 35, [0, inf], "volume", |
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[deb7ee0] | 103 | "Shorter side of the parallelepiped"], |
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| 104 | ["b2a_ratio", "Ang", 1, [0, inf], "volume", |
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| 105 | "Ratio sides b/a"], |
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| 106 | ["c2a_ratio", "Ang", 1, [0, inf], "volume", |
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| 107 | "Ratio sides c/a"], |
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| 108 | ] |
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| 109 | |
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[3d8283b] | 110 | source = ["lib/gauss76.c", "hollow_rectangular_prism_thin_walls.c"] |
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[71b751d] | 111 | have_Fq = True |
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[ee60aa7] | 112 | effective_radius_type = [ |
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[99658f6] | 113 | "equivalent cylinder excluded volume", "equivalent outer volume sphere", |
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| 114 | "half length_a", "half length_b", "half length_c", |
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[ee60aa7] | 115 | "equivalent outer circular cross-section", |
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| 116 | "half ab diagonal", "half diagonal", |
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| 117 | ] |
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[deb7ee0] | 118 | |
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| 119 | |
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[31df0c9] | 120 | def random(): |
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| 121 | a, b, c = 10**np.random.uniform(1, 4.7, size=3) |
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| 122 | pars = dict( |
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| 123 | length_a=a, |
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| 124 | b2a_ratio=b/a, |
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| 125 | c2a_ratio=c/a, |
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| 126 | ) |
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| 127 | return pars |
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| 128 | |
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| 129 | |
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[deb7ee0] | 130 | # parameters for demo |
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| 131 | demo = dict(scale=1, background=0, |
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[ab2aea8] | 132 | sld=6.3, sld_solvent=1.0, |
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[a807206] | 133 | length_a=35, b2a_ratio=1, c2a_ratio=1, |
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| 134 | length_a_pd=0.1, length_a_pd_n=10, |
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[deb7ee0] | 135 | b2a_ratio_pd=0.1, b2a_ratio_pd_n=1, |
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| 136 | c2a_ratio_pd=0.1, c2a_ratio_pd_n=1) |
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| 137 | |
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[6dd90c1] | 138 | tests = [[{}, 0.2, 0.837719188592], |
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| 139 | [{}, [0.2], [0.837719188592]], |
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[deb7ee0] | 140 | ] |
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