[81dd619] | 1 | r""" |
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| 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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[b99734a] | 5 | Parameters for this model are the core axial ratio X and a shell thickness, |
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| 6 | which are more often what we would like to determine and makes the model |
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| 7 | better behaved, particularly when polydispersity is applied than the four |
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| 8 | independent radii used in the original parameterization of this model. |
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| 9 | |
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| 10 | |
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[5031ca3] | 11 | .. figure:: img/core_shell_ellipsoid_geometry.png |
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[81dd619] | 12 | |
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[9272cbd] | 13 | The geometric parameters of this model are shown in the diagram above, which |
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[416f5c7] | 14 | shows (a) a cut through at the circular equator and (b) a cross section through |
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[9272cbd] | 15 | the poles, of a prolate ellipsoid. |
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[81dd619] | 16 | |
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[5031ca3] | 17 | When *X_core < 1* the core is oblate; when *X_core > 1* it is prolate. |
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| 18 | *X_core = 1* is a spherical core. |
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[81dd619] | 19 | |
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[5031ca3] | 20 | For a fixed shell thickness *XpolarShell = 1*, to scale the shell thickness |
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[416f5c7] | 21 | pro-rata with the radius set or constrain *XpolarShell = X_core*. |
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[81dd619] | 22 | |
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[5031ca3] | 23 | When including an $S(q)$, the radius in $S(q)$ is calculated to be that of |
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| 24 | a sphere with the same 2nd virial coefficient of the outer surface of the |
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| 25 | ellipsoid. This may have some undesirable effects if the aspect ratio of the |
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| 26 | ellipsoid is large (ie, if $X << 1$ or $X >> 1$ ), when the $S(q)$ |
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[2a0b2b1] | 27 | - which assumes spheres - will not in any case be valid. Generating a |
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[2d81cfe] | 28 | custom product model will enable separate effective volume fraction and |
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| 29 | effective radius in the $S(q)$. |
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[81dd619] | 30 | |
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[5031ca3] | 31 | If SAS data are in absolute units, and the SLDs are correct, then scale should |
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| 32 | be the total volume fraction of the "outer particle". When $S(q)$ is introduced |
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[2d81cfe] | 33 | this moves to the $S(q)$ volume fraction, and scale should then be 1.0, or |
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| 34 | contain some other units conversion factor (for example, if you have SAXS data). |
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[81dd619] | 35 | |
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[2d81cfe] | 36 | The calculation of intensity follows that for the solid ellipsoid, but |
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| 37 | with separate terms for the core-shell and shell-solvent boundaries. |
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[416f5c7] | 38 | |
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| 39 | .. math:: |
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| 40 | |
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| 41 | P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \text{background} |
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| 42 | |
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| 43 | where |
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| 44 | |
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| 45 | .. math:: |
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[2e0c0b0] | 46 | :nowrap: |
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[17fb550] | 47 | |
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[30b60d2] | 48 | \begin{align*} |
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[416f5c7] | 49 | F(q,\alpha) = &f(q,radius\_equat\_core,radius\_equat\_core.x\_core,\alpha) \\ |
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[2d81cfe] | 50 | &+ f(q,radius\_equat\_core + thick\_shell, |
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| 51 | radius\_equat\_core.x\_core + thick\_shell.x\_polar\_shell,\alpha) |
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[30b60d2] | 52 | \end{align*} |
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[416f5c7] | 53 | |
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| 54 | where |
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[2a0b2b1] | 55 | |
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[416f5c7] | 56 | .. math:: |
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| 57 | |
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| 58 | f(q,R_e,R_p,\alpha) = \frac{3 \Delta \rho V (\sin[qr(R_p,R_e,\alpha)] |
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| 59 | - \cos[qr(R_p,R_e,\alpha)])} |
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| 60 | {[qr(R_p,R_e,\alpha)]^3} |
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| 61 | |
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| 62 | and |
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| 63 | |
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| 64 | .. math:: |
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| 65 | |
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| 66 | r(R_e,R_p,\alpha) = \left[ R_e^2 \sin^2 \alpha |
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| 67 | + R_p^2 \cos^2 \alpha \right]^{1/2} |
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| 68 | |
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| 69 | |
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| 70 | $\alpha$ is the angle between the axis of the ellipsoid and $\vec q$, |
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[2d81cfe] | 71 | $V = (4/3)\pi R_pR_e^2$ is the volume of the ellipsoid , $R_p$ is the |
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| 72 | polar radius along the rotational axis of the ellipsoid, $R_e$ is the |
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| 73 | equatorial radius perpendicular to the rotational axis of the ellipsoid |
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| 74 | and $\Delta \rho$ (contrast) is the scattering length density difference, |
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| 75 | either $(sld\_core - sld\_shell)$ or $(sld\_shell - sld\_solvent)$. |
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[416f5c7] | 76 | |
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| 77 | For randomly oriented particles: |
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| 78 | |
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| 79 | .. math:: |
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| 80 | |
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| 81 | F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha} |
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| 82 | |
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[2d81cfe] | 83 | For oriented ellipsoids the *theta*, *phi* and *psi* orientation parameters |
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| 84 | will appear when fitting 2D data, see the :ref:`elliptical-cylinder` model |
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| 85 | for further information. |
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[416f5c7] | 86 | |
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[81dd619] | 87 | References |
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| 88 | ---------- |
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[9272cbd] | 89 | see for example: |
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| 90 | Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461. |
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| 91 | Berr, S. J. Phys. Chem., 1987, 91, 4760. |
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| 92 | |
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| 93 | Authorship and Verification |
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| 94 | ---------------------------- |
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[81dd619] | 95 | |
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[9272cbd] | 96 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
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| 97 | * **Last Modified by:** Richard Heenan (reparametrised model) **Date:** 2015 |
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| 98 | * **Last Reviewed by:** Richard Heenan **Date:** October 6, 2016 |
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[81dd619] | 99 | """ |
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| 100 | |
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[2d81cfe] | 101 | import numpy as np |
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[81dd619] | 102 | from numpy import inf, sin, cos, pi |
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| 103 | |
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[b99734a] | 104 | name = "core_shell_ellipsoid" |
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[81dd619] | 105 | title = "Form factor for an spheroid ellipsoid particle with a core shell structure." |
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| 106 | description = """ |
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[b99734a] | 107 | [core_shell_ellipsoid] Calculates the form factor for an spheroid |
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[5031ca3] | 108 | ellipsoid particle with a core_shell structure. |
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| 109 | The form factor is averaged over all possible |
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| 110 | orientations of the ellipsoid such that P(q) |
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| 111 | = scale*<f^2>/Vol + bkg, where f is the |
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| 112 | single particle scattering amplitude. |
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| 113 | [Parameters]: |
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| 114 | radius_equat_core = equatorial radius of core, |
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| 115 | x_core = ratio of core polar/equatorial radii, |
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| 116 | thick_shell = equatorial radius of outer surface, |
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| 117 | x_polar_shell = ratio of polar shell thickness to equatorial shell thickness, |
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| 118 | sld_core = SLD_core |
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| 119 | sld_shell = SLD_shell |
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| 120 | sld_solvent = SLD_solvent |
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| 121 | background = Incoherent bkg |
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| 122 | scale =scale |
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| 123 | Note:It is the users' responsibility to ensure |
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| 124 | that shell radii are larger than core radii. |
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| 125 | oblate: polar radius < equatorial radius |
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| 126 | prolate : polar radius > equatorial radius - this new model will make this easier |
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| 127 | and polydispersity integrals more logical (as previously the shell could disappear). |
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[81dd619] | 128 | """ |
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| 129 | category = "shape:ellipsoid" |
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| 130 | |
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| 131 | # pylint: disable=bad-whitespace, line-too-long |
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[5031ca3] | 132 | # ["name", "units", default, [lower, upper], "type", "description"], |
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[81dd619] | 133 | parameters = [ |
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[73e08ae] | 134 | ["radius_equat_core","Ang", 20, [0, inf], "volume", "Equatorial radius of core"], |
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[5031ca3] | 135 | ["x_core", "None", 3, [0, inf], "volume", "axial ratio of core, X = r_polar/r_equatorial"], |
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| 136 | ["thick_shell", "Ang", 30, [0, inf], "volume", "thickness of shell at equator"], |
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| 137 | ["x_polar_shell", "", 1, [0, inf], "volume", "ratio of thickness of shell at pole to that at equator"], |
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| 138 | ["sld_core", "1e-6/Ang^2", 2, [-inf, inf], "sld", "Core scattering length density"], |
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| 139 | ["sld_shell", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Shell scattering length density"], |
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| 140 | ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"], |
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[9b79f29] | 141 | ["theta", "degrees", 0, [-360, 360], "orientation", "elipsoid axis to beam angle"], |
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| 142 | ["phi", "degrees", 0, [-360, 360], "orientation", "rotation about beam"], |
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[81dd619] | 143 | ] |
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| 144 | # pylint: enable=bad-whitespace, line-too-long |
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| 145 | |
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[2a0b2b1] | 146 | source = ["lib/sas_3j1x_x.c", "lib/gauss76.c", "core_shell_ellipsoid.c"] |
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[71b751d] | 147 | have_Fq = True |
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[d277229] | 148 | effective_radius_type = ["equivalent sphere","average outer curvature", "min outer radius", "max outer radius"] |
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| 149 | |
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| 150 | #def ER(radius_equat_core, x_core, thick_shell, x_polar_shell): |
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| 151 | # """ |
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| 152 | # Returns the effective radius used in the S*P calculation |
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| 153 | # """ |
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| 154 | # from .ellipsoid import ER as ellipsoid_ER |
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| 155 | # polar_outer = radius_equat_core*x_core + thick_shell*x_polar_shell |
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| 156 | # equat_outer = radius_equat_core + thick_shell |
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| 157 | # return ellipsoid_ER(polar_outer, equat_outer) |
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[65bf704] | 158 | |
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[31df0c9] | 159 | def random(): |
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[2d81cfe] | 160 | volume = 10**np.random.uniform(5, 12) |
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[8f04da4] | 161 | outer_polar = 10**np.random.uniform(1.3, 4) |
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[2d81cfe] | 162 | outer_equatorial = np.sqrt(volume/outer_polar) # ignore 4/3 pi |
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[8f04da4] | 163 | # Use a distribution with a preference for thin shell or thin core |
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| 164 | # Avoid core,shell radii < 1 |
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[9f6823b] | 165 | thickness_polar = np.random.beta(0.5, 0.5)*(outer_polar-2) + 1 |
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[8f04da4] | 166 | thickness_equatorial = np.random.beta(0.5, 0.5)*(outer_equatorial-2) + 1 |
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| 167 | radius_polar = outer_polar - thickness_polar |
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| 168 | radius_equatorial = outer_equatorial - thickness_equatorial |
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[31df0c9] | 169 | x_core = radius_polar/radius_equatorial |
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| 170 | x_polar_shell = thickness_polar/thickness_equatorial |
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| 171 | pars = dict( |
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| 172 | #background=0, sld=0, sld_solvent=1, |
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| 173 | radius_equat_core=radius_equatorial, |
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| 174 | x_core=x_core, |
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| 175 | thick_shell=thickness_equatorial, |
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| 176 | x_polar_shell=x_polar_shell, |
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| 177 | ) |
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| 178 | return pars |
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[81dd619] | 179 | |
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| 180 | q = 0.1 |
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[b7e8b94] | 181 | # tests had in old coords theta=0, phi=0; new coords theta=90, phi=0 |
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| 182 | qx = q*cos(pi/6.0) |
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| 183 | qy = q*sin(pi/6.0) |
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| 184 | # 11Jan2017 RKH sorted tests after redefinition of angles |
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[fcb33e4] | 185 | tests = [ |
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[2d81cfe] | 186 | # Accuracy tests based on content in test/utest_coreshellellipsoidXTmodel.py |
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[fcb33e4] | 187 | [{'radius_equat_core': 200.0, |
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| 188 | 'x_core': 0.1, |
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| 189 | 'thick_shell': 50.0, |
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| 190 | 'x_polar_shell': 0.2, |
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| 191 | 'sld_core': 2.0, |
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| 192 | 'sld_shell': 1.0, |
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| 193 | 'sld_solvent': 6.3, |
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| 194 | 'background': 0.001, |
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| 195 | 'scale': 1.0, |
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| 196 | }, 1.0, 0.00189402], |
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[81dd619] | 197 | |
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| 198 | # Additional tests with larger range of parameters |
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[fcb33e4] | 199 | [{'background': 0.01}, 0.1, 11.6915], |
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| 200 | |
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| 201 | [{'radius_equat_core': 20.0, |
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| 202 | 'x_core': 200.0, |
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| 203 | 'thick_shell': 54.0, |
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| 204 | 'x_polar_shell': 3.0, |
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| 205 | 'sld_core': 20.0, |
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| 206 | 'sld_shell': 10.0, |
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| 207 | 'sld_solvent': 6.0, |
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| 208 | 'background': 0.0, |
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| 209 | 'scale': 1.0, |
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| 210 | }, 0.01, 8688.53], |
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[7c2935c] | 211 | |
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[2d81cfe] | 212 | # 2D tests |
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| 213 | [{'background': 0.001, |
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| 214 | 'theta': 90.0, |
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| 215 | 'phi': 0.0, |
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[7c2935c] | 216 | }, (0.4, 0.5), 0.00690673], |
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[fcb33e4] | 217 | |
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[2d81cfe] | 218 | [{'radius_equat_core': 20.0, |
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[fcb33e4] | 219 | 'x_core': 200.0, |
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| 220 | 'thick_shell': 54.0, |
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| 221 | 'x_polar_shell': 3.0, |
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| 222 | 'sld_core': 20.0, |
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| 223 | 'sld_shell': 10.0, |
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| 224 | 'sld_solvent': 6.0, |
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| 225 | 'background': 0.01, |
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| 226 | 'scale': 0.01, |
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[b7e8b94] | 227 | 'theta': 90.0, |
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[7c2935c] | 228 | 'phi': 0.0, |
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[fcb33e4] | 229 | }, (qx, qy), 0.01000025], |
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[2d81cfe] | 230 | ] |
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