source: sasmodels/sasmodels/models/core_shell_ellipsoid.py @ e5a8f33

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
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1r"""
2Definition
3----------
4
5Parameters for this model are the core axial ratio $X_{core}$ and a shell
6thickness $t_{shell}$, which are more often what we would like to determine
7and make the model better behaved, particularly when polydispersity is
8applied, than the four independent radii used in the original parameterization
9of this model.
10
11
12.. figure:: img/core_shell_ellipsoid_geometry.png
13
14The geometric parameters of this model are shown in the diagram above, which
15shows (a) a cut through at the circular equator and (b) a cross section through
16the poles, of a prolate ellipsoid.
17
18When $X_{core}$ < 1 the core is oblate; when $X_{core}$ > 1 it is prolate.
19$X_{core}$ = 1 is a spherical core.
20
21For a fixed shell thickness $X_{polar shell}$ = 1, to scale $t_{shell}$
22pro-rata with the radius set or constrain $X_{polar shell}$ = $X_{core}$.
23
24.. note::
25
26   When including an $S(q)$, the radius in $S(q)$ is calculated to be that of
27   a sphere with the same 2nd virial coefficient of the outer surface of the
28   ellipsoid. This may have some undesirable effects if the aspect ratio of the
29   ellipsoid is large (ie, if $X << 1$ or $X >> 1$), when the $S(q)$
30   - which assumes spheres - will not in any case be valid.  Generating a
31   custom product model will enable separate effective volume fraction and
32   effective radius in the $S(q)$.
33
34If SAS data are in absolute units, and the SLDs are correct, then scale should
35be the total volume fraction of the "outer particle". When $S(q)$ is introduced
36this moves to the $S(q)$ volume fraction, and scale should then be 1.0, or
37contain some other units conversion factor (for example, if you have SAXS data).
38
39The calculation of intensity follows that for the solid ellipsoid, but
40with separate terms for the core-shell and shell-solvent boundaries.
41
42.. math::
43
44    P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \text{background}
45
46where
47
48.. In following equation SK changed radius\_equat\_core to R_e
49 
50.. math::
51    :nowrap:
52
53    \begin{align*}
54    F(q,\alpha) = &f(q,R_e,R_e.x_{core},\alpha) \\
55    &+ f(q,R_e + t_{shell},
56         R_e.x_{core} + t_{shell}.x_{polar shell},\alpha)
57    \end{align*}
58
59where
60
61.. math::
62
63    f(q,R_e,R_p,\alpha) = \frac{3 \Delta \rho V (\sin[qr(R_p,R_e,\alpha)]
64                - \cos[qr(R_p,R_e,\alpha)])}
65                {[qr(R_p,R_e,\alpha)]^3}
66
67and
68
69.. math::
70
71    r(R_e,R_p,\alpha) = \left[ R_e^2 \sin^2 \alpha
72        + R_p^2 \cos^2 \alpha \right]^{1/2}
73
74
75$\alpha$ is the angle between the axis of the ellipsoid and $\vec q$,
76$V = (4/3)\pi R_pR_e^2$ is the volume of the ellipsoid , $R_p$ is the
77polar radius along the rotational axis of the ellipsoid, $R_e$ is the
78equatorial radius perpendicular to the rotational axis of the ellipsoid,
79$t_{shell}$ is the thickness of the shell at the equator,
80and $\Delta \rho$ (the contrast) is the scattering length density difference,
81either $(\rho_{core} - \rho_{shell})$ or $(\rho_{shell} - \rho_{solvent})$.
82
83For randomly oriented particles:
84
85.. math::
86
87   F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}
88
89For oriented ellipsoids the *theta*, *phi* and *psi* orientation parameters
90will appear when fitting 2D data, see the :ref:`elliptical-cylinder` model
91for further information.
92
93References
94----------
95see for example:
96
97.. [#] Kotlarchyk, M.; Chen, S.-H. *J. Chem. Phys.*, 1983, 79, 2461
98.. [#] Berr, S. *J. Phys. Chem.*, 1987, 91, 4760
99
100Source
101------
102
103`core_shell_ellipsoid.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/core_shell_ellipsoid.py>`_
104
105`core_shell_ellipsoid.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/core_shell_ellipsoid.c>`_
106
107Authorship and Verification
108----------------------------
109
110* **Author:** NIST IGOR/DANSE **Date:** pre 2010
111* **Last Modified by:** Richard Heenan (reparametrised model) **Date:** 2015
112* **Last Reviewed by:** Steve King **Date:** March 27, 2019
113* **Source added by :** Steve King **Date:** March 25, 2019
114"""
115
116import numpy as np
117from numpy import inf, sin, cos, pi
118
119name = "core_shell_ellipsoid"
120title = "Form factor for an spheroid ellipsoid particle with a core shell structure."
121description = """
122        [core_shell_ellipsoid] Calculates the form factor for an spheroid
123        ellipsoid particle with a core_shell structure.
124        The form factor is averaged over all possible
125        orientations of the ellipsoid such that P(q)
126        = scale*<f^2>/Vol + bkg, where f is the
127        single particle scattering amplitude.
128        [Parameters]:
129        radius_equat_core = equatorial radius of core,
130        x_core = ratio of core polar/equatorial radii,
131        thick_shell = equatorial radius of outer surface,
132        x_polar_shell = ratio of polar shell thickness to equatorial shell thickness,
133        sld_core = SLD_core
134        sld_shell = SLD_shell
135        sld_solvent = SLD_solvent
136        background = Incoherent bkg
137        scale =scale
138        Note:It is the users' responsibility to ensure
139        that shell radii are larger than core radii.
140        oblate: polar radius < equatorial radius
141        prolate :  polar radius > equatorial radius - this new model will make this easier
142        and polydispersity integrals more logical (as previously the shell could disappear).
143    """
144category = "shape:ellipsoid"
145
146# pylint: disable=bad-whitespace, line-too-long
147#             ["name", "units", default, [lower, upper], "type", "description"],
148parameters = [
149    ["radius_equat_core","Ang",     20,   [0, inf],   "volume",      "Equatorial radius of core"],
150    ["x_core",        "None",       3,   [0, inf],    "volume",      "axial ratio of core, X = r_polar/r_equatorial"],
151    ["thick_shell",   "Ang",       30,   [0, inf],    "volume",      "thickness of shell at equator"],
152    ["x_polar_shell", "",           1,   [0, inf],    "volume",      "ratio of thickness of shell at pole to that at equator"],
153    ["sld_core",      "1e-6/Ang^2", 2,   [-inf, inf], "sld",         "Core scattering length density"],
154    ["sld_shell",     "1e-6/Ang^2", 1,   [-inf, inf], "sld",         "Shell scattering length density"],
155    ["sld_solvent",   "1e-6/Ang^2", 6.3, [-inf, inf], "sld",         "Solvent scattering length density"],
156    ["theta",         "degrees",    0,   [-360, 360], "orientation", "elipsoid axis to beam angle"],
157    ["phi",           "degrees",    0,   [-360, 360], "orientation", "rotation about beam"],
158    ]
159# pylint: enable=bad-whitespace, line-too-long
160
161source = ["lib/sas_3j1x_x.c", "lib/gauss76.c", "core_shell_ellipsoid.c"]
162have_Fq = True
163effective_radius_type = [
164    "average outer curvature", "equivalent volume sphere",
165    "min outer radius", "max outer radius",
166    ]
167
168def random():
169    """Return a random parameter set for the model."""
170    volume = 10**np.random.uniform(5, 12)
171    outer_polar = 10**np.random.uniform(1.3, 4)
172    outer_equatorial = np.sqrt(volume/outer_polar) # ignore 4/3 pi
173    # Use a distribution with a preference for thin shell or thin core
174    # Avoid core,shell radii < 1
175    thickness_polar = np.random.beta(0.5, 0.5)*(outer_polar-2) + 1
176    thickness_equatorial = np.random.beta(0.5, 0.5)*(outer_equatorial-2) + 1
177    radius_polar = outer_polar - thickness_polar
178    radius_equatorial = outer_equatorial - thickness_equatorial
179    x_core = radius_polar/radius_equatorial
180    x_polar_shell = thickness_polar/thickness_equatorial
181    pars = dict(
182        #background=0, sld=0, sld_solvent=1,
183        radius_equat_core=radius_equatorial,
184        x_core=x_core,
185        thick_shell=thickness_equatorial,
186        x_polar_shell=x_polar_shell,
187    )
188    return pars
189
190q = 0.1
191# tests had in old coords theta=0, phi=0; new coords theta=90, phi=0
192qx = q*cos(pi/6.0)
193qy = q*sin(pi/6.0)
194# 11Jan2017 RKH sorted tests after redefinition of angles
195tests = [
196    # Accuracy tests based on content in test/utest_coreshellellipsoidXTmodel.py
197    [{'radius_equat_core': 200.0,
198      'x_core': 0.1,
199      'thick_shell': 50.0,
200      'x_polar_shell': 0.2,
201      'sld_core': 2.0,
202      'sld_shell': 1.0,
203      'sld_solvent': 6.3,
204      'background': 0.001,
205      'scale': 1.0,
206     }, 1.0, 0.00189402],
207
208    # Additional tests with larger range of parameters
209    [{'background': 0.01}, 0.1, 11.6915],
210
211    [{'radius_equat_core': 20.0,
212      'x_core': 200.0,
213      'thick_shell': 54.0,
214      'x_polar_shell': 3.0,
215      'sld_core': 20.0,
216      'sld_shell': 10.0,
217      'sld_solvent': 6.0,
218      'background': 0.0,
219      'scale': 1.0,
220     }, 0.01, 8688.53],
221
222    # 2D tests
223    [{'background': 0.001,
224      'theta': 90.0,
225      'phi': 0.0,
226     }, (0.4, 0.5), 0.00690673],
227
228    [{'radius_equat_core': 20.0,
229      'x_core': 200.0,
230      'thick_shell': 54.0,
231      'x_polar_shell': 3.0,
232      'sld_core': 20.0,
233      'sld_shell': 10.0,
234      'sld_solvent': 6.0,
235      'background': 0.01,
236      'scale': 0.01,
237      'theta': 90.0,
238      'phi': 0.0,
239     }, (qx, qy), 0.01000025],
240]
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