[fcb33e4] | 1 | r""" |
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| 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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| 5 | This model provides the form factor for an elliptical cylinder with a |
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| 6 | core-shell scattering length density profile. Thus this is a variation |
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| 7 | of the core-shell bicelle model, but with an elliptical cylinder for the core. |
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[8f04da4] | 8 | Outer shells on the rims and flat ends may be of different thicknesses and |
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[2d81cfe] | 9 | scattering length densities. The form factor is normalized by the total |
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| 10 | particle volume. |
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[fcb33e4] | 11 | |
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| 12 | .. figure:: img/core_shell_bicelle_geometry.png |
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| 13 | |
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| 14 | Schematic cross-section of bicelle. Note however that the model here |
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| 15 | calculates for rectangular, not curved, rims as shown below. |
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| 16 | |
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| 17 | .. figure:: img/core_shell_bicelle_parameters.png |
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| 18 | |
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[8f04da4] | 19 | Cross section of model used here. Users will have |
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| 20 | to decide how to distribute "heads" and "tails" between the rim, face |
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[fcb33e4] | 21 | and core regions in order to estimate appropriate starting parameters. |
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| 22 | |
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| 23 | Given the scattering length densities (sld) $\rho_c$, the core sld, $\rho_f$, |
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| 24 | the face sld, $\rho_r$, the rim sld and $\rho_s$ the solvent sld, the |
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| 25 | scattering length density variation along the bicelle axis is: |
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| 26 | |
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| 27 | .. math:: |
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| 28 | |
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[8f04da4] | 29 | \rho(r) = |
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| 30 | \begin{cases} |
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[fcb33e4] | 31 | &\rho_c \text{ for } 0 \lt r \lt R; -L \lt z\lt L \\[1.5ex] |
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| 32 | &\rho_f \text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; |
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| 33 | L \lt z\lt (L+2t) \\[1.5ex] |
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| 34 | &\rho_r\text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t) |
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| 35 | \end{cases} |
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| 36 | |
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| 37 | The form factor for the bicelle is calculated in cylindrical coordinates, where |
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[2d81cfe] | 38 | $\alpha$ is the angle between the $Q$ vector and the cylinder axis, and $\psi$ |
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| 39 | is the angle for the ellipsoidal cross section core, to give: |
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[fcb33e4] | 40 | |
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| 41 | .. math:: |
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| 42 | |
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| 43 | I(Q,\alpha,\psi) = \frac{\text{scale}}{V_t} \cdot |
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[a53bf6b] | 44 | F(Q,\alpha, \psi)^2 \cdot sin(\alpha) + \text{background} |
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[fcb33e4] | 45 | |
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[2d81cfe] | 46 | where a numerical integration of $F(Q,\alpha, \psi)^2 \cdot sin(\alpha)$ |
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| 47 | is carried out over \alpha and \psi for: |
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[fcb33e4] | 48 | |
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| 49 | .. math:: |
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[17fb550] | 50 | :nowrap: |
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| 51 | |
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[30b60d2] | 52 | \begin{align*} |
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[8f04da4] | 53 | F(Q,\alpha,\psi) = &\bigg[ |
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[2d81cfe] | 54 | (\rho_c - \rho_f) V_c |
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| 55 | \frac{2J_1(QR'sin \alpha)}{QR'sin\alpha} |
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| 56 | \frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\ |
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| 57 | &+(\rho_f - \rho_r) V_{c+f} |
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| 58 | \frac{2J_1(QR'sin\alpha)}{QR'sin\alpha} |
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| 59 | \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} \\ |
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| 60 | &+(\rho_r - \rho_s) V_t |
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| 61 | \frac{2J_1(Q(R'+t_r)sin\alpha)}{Q(R'+t_r)sin\alpha} |
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| 62 | \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} |
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[fcb33e4] | 63 | \bigg] |
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[30b60d2] | 64 | \end{align*} |
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[fcb33e4] | 65 | |
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| 66 | where |
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| 67 | |
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| 68 | .. math:: |
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| 69 | |
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| 70 | R'=\frac{R}{\sqrt{2}}\sqrt{(1+X_{core}^{2}) + (1-X_{core}^{2})cos(\psi)} |
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[8f04da4] | 71 | |
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| 72 | |
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[2d81cfe] | 73 | and $V_t = \pi.(R+t_r)(Xcore.R+t_r)^2.(L+2.t_f)$ is the total volume of |
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| 74 | the bicelle, $V_c = \pi.Xcore.R^2.L$ the volume of the core, |
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| 75 | $V_{c+f} = \pi.Xcore.R^2.(L+2.t_f)$ the volume of the core plus the volume |
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| 76 | of the faces, $R$ is the radius of the core, $Xcore$ is the axial ratio of |
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| 77 | the core, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$ |
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| 78 | the thickness of the rim and $J_1$ the usual first order bessel function. |
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| 79 | The core has radii $R$ and $Xcore.R$ so is circular, as for the |
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| 80 | core_shell_bicelle model, for $Xcore$ =1. Note that you may need to |
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| 81 | limit the range of $Xcore$, especially if using the Monte-Carlo algorithm, |
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| 82 | as setting radius to $R/Xcore$ and axial ratio to $1/Xcore$ gives an |
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| 83 | equivalent solution! |
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[fcb33e4] | 84 | |
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| 85 | The output of the 1D scattering intensity function for randomly oriented |
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| 86 | bicelles is then given by integrating over all possible $\alpha$ and $\psi$. |
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| 87 | |
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[2d81cfe] | 88 | For oriented bicelles the *theta*, *phi* and *psi* orientation parameters will |
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| 89 | appear when fitting 2D data, see the :ref:`elliptical-cylinder` model |
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| 90 | for further information. |
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[fcb33e4] | 91 | |
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[15a90c1] | 92 | .. figure:: img/elliptical_cylinder_angle_definition.png |
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| 93 | |
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[8f04da4] | 94 | Definition of the angles for the oriented core_shell_bicelle_elliptical particles. |
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[fcb33e4] | 95 | |
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[fc3ae1b] | 96 | Model verified using Monte Carlo simulation for 1D and 2D scattering. |
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[fcb33e4] | 97 | |
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| 98 | References |
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| 99 | ---------- |
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| 100 | |
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| 101 | .. [#] |
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[b297ba9] | 102 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
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[fcb33e4] | 103 | |
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| 104 | Authorship and Verification |
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| 105 | ---------------------------- |
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| 106 | |
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| 107 | * **Author:** Richard Heenan **Date:** December 14, 2016 |
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| 108 | * **Last Modified by:** Richard Heenan **Date:** December 14, 2016 |
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[fc3ae1b] | 109 | * **Last Reviewed by:** Paul Kienzle **Date:** Feb 28, 2018 |
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[fcb33e4] | 110 | """ |
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| 111 | |
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[2d81cfe] | 112 | import numpy as np |
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[0b56f38] | 113 | from numpy import inf, sin, cos, pi |
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[fcb33e4] | 114 | |
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| 115 | name = "core_shell_bicelle_elliptical" |
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| 116 | title = "Elliptical cylinder with a core-shell scattering length density profile.." |
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| 117 | description = """ |
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| 118 | core_shell_bicelle_elliptical |
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[8f04da4] | 119 | Elliptical cylinder core, optional shell on the two flat faces, and shell of |
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| 120 | uniform thickness on its rim (extending around the end faces). |
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[fcb33e4] | 121 | Please see full documentation for equations and further details. |
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| 122 | Involves a double numerical integral around the ellipsoid diameter |
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| 123 | and the angle of the cylinder axis to Q. |
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| 124 | Compare also the core_shell_bicelle and elliptical_cylinder models. |
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| 125 | """ |
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| 126 | category = "shape:cylinder" |
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| 127 | |
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| 128 | # pylint: disable=bad-whitespace, line-too-long |
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| 129 | # ["name", "units", default, [lower, upper], "type", "description"], |
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| 130 | parameters = [ |
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[fc3ae1b] | 131 | ["radius", "Ang", 30, [0, inf], "volume", "Cylinder core radius r_minor"], |
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| 132 | ["x_core", "None", 3, [0, inf], "volume", "Axial ratio of core, X = r_major/r_minor"], |
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[9b79f29] | 133 | ["thick_rim", "Ang", 8, [0, inf], "volume", "Rim shell thickness"], |
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| 134 | ["thick_face", "Ang", 14, [0, inf], "volume", "Cylinder face thickness"], |
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| 135 | ["length", "Ang", 50, [0, inf], "volume", "Cylinder length"], |
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[fcb33e4] | 136 | ["sld_core", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Cylinder core scattering length density"], |
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| 137 | ["sld_face", "1e-6/Ang^2", 7, [-inf, inf], "sld", "Cylinder face scattering length density"], |
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| 138 | ["sld_rim", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Cylinder rim scattering length density"], |
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| 139 | ["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld", "Solvent scattering length density"], |
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[fc3ae1b] | 140 | ["theta", "degrees", 90.0, [-360, 360], "orientation", "Cylinder axis to beam angle"], |
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| 141 | ["phi", "degrees", 0, [-360, 360], "orientation", "Rotation about beam"], |
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| 142 | ["psi", "degrees", 0, [-360, 360], "orientation", "Rotation about cylinder axis"] |
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[fcb33e4] | 143 | ] |
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| 144 | |
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| 145 | # pylint: enable=bad-whitespace, line-too-long |
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| 146 | |
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[4b541ac] | 147 | source = ["lib/sas_Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", |
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| 148 | "core_shell_bicelle_elliptical.c"] |
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[71b751d] | 149 | have_Fq = True |
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[ee60aa7] | 150 | effective_radius_type = [ |
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[b297ba9] | 151 | "equivalent cylinder excluded volume", "equivalent volume sphere", |
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| 152 | "outer rim average radius", "outer rim min radius", |
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[ee60aa7] | 153 | "outer max radius", "half outer thickness", "half diagonal", |
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| 154 | ] |
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[fcb33e4] | 155 | |
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[8f04da4] | 156 | def random(): |
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[b297ba9] | 157 | """Return a random parameter set for the model.""" |
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[8f04da4] | 158 | outer_major = 10**np.random.uniform(1, 4.7) |
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| 159 | outer_minor = 10**np.random.uniform(1, 4.7) |
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| 160 | # Use a distribution with a preference for thin shell or thin core, |
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| 161 | # limited by the minimum radius. Avoid core,shell radii < 1 |
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| 162 | min_radius = min(outer_major, outer_minor) |
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| 163 | thick_rim = np.random.beta(0.5, 0.5)*(min_radius-2) + 1 |
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| 164 | radius_major = outer_major - thick_rim |
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| 165 | radius_minor = outer_minor - thick_rim |
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| 166 | radius = radius_major |
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| 167 | x_core = radius_minor/radius_major |
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| 168 | outer_length = 10**np.random.uniform(1, 4.7) |
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| 169 | # Caps should be a small percentage of the total length, but at least one |
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| 170 | # angstrom long. Since outer length >= 10, the following will suffice |
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| 171 | thick_face = 10**np.random.uniform(-np.log10(outer_length), -1)*outer_length |
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| 172 | length = outer_length - thick_face |
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| 173 | pars = dict( |
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| 174 | radius=radius, |
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| 175 | x_core=x_core, |
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| 176 | thick_rim=thick_rim, |
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| 177 | thick_face=thick_face, |
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| 178 | length=length |
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| 179 | ) |
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| 180 | return pars |
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| 181 | |
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[fcb33e4] | 182 | |
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[0b56f38] | 183 | q = 0.1 |
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| 184 | # april 6 2017, rkh added a 2d unit test, NOT READY YET pull #890 branch assume correct! |
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| 185 | qx = q*cos(pi/6.0) |
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| 186 | qy = q*sin(pi/6.0) |
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[fcb33e4] | 187 | |
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| 188 | tests = [ |
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[304c775] | 189 | #[{'radius': 30.0, 'x_core': 3.0, |
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| 190 | # 'thick_rim': 8.0, 'thick_face': 14.0, 'length': 50.0}, 'ER', 1], |
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| 191 | #[{'radius': 30.0, 'x_core': 3.0, |
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| 192 | # 'thick_rim': 8.0, 'thick_face': 14.0, 'length': 50.0}, 'VR', 1], |
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[2d81cfe] | 193 | |
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| 194 | [{'radius': 30.0, 'x_core': 3.0, |
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| 195 | 'thick_rim': 8.0, 'thick_face': 14.0, 'length': 50.0, |
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| 196 | 'sld_core': 4.0, 'sld_face': 7.0, 'sld_rim': 1.0, |
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| 197 | 'sld_solvent': 6.0, 'background': 0.0}, |
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| 198 | 0.015, 286.540286], |
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| 199 | #[{'theta':80., 'phi':10.}, (qx, qy), 7.88866563001], |
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| 200 | ] |
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[0b56f38] | 201 | |
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| 202 | del qx, qy # not necessary to delete, but cleaner |
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