source: sasmodels/sasmodels/models/core_shell_ellipsoid.py @ 71b751d

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 71b751d was 71b751d, checked in by Paul Kienzle <pkienzle@…>, 23 months ago

update remaining form factors to use Fq interface

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