source: sasmodels/sasmodels/models/core_shell_ellipsoid.py @ 2a0b2b1

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 2a0b2b1 was 2a0b2b1, checked in by Paul Kienzle <pkienzle@…>, 7 years ago

restructure all 2D models to work with (qa,qb,qc) = rotate(qx,qy) rather than working with angles directly in preparation for revised jitter algorithm

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File size: 7.8 KB
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 effective
29radius 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,
34or contain some other units conversion factor (for example, if you have SAXS data).
35
36The calculation of intensity follows that for the solid ellipsoid, but with separate
37terms 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    \begin{align}
47    F(q,\alpha) = &f(q,radius\_equat\_core,radius\_equat\_core.x\_core,\alpha) \\
48    &+ f(q,radius\_equat\_core + thick\_shell,radius\_equat\_core.x\_core + thick\_shell.x\_polar\_shell,\alpha)
49    \end{align}
50
51where
52
53.. math::
54
55    f(q,R_e,R_p,\alpha) = \frac{3 \Delta \rho V (\sin[qr(R_p,R_e,\alpha)]
56                - \cos[qr(R_p,R_e,\alpha)])}
57                {[qr(R_p,R_e,\alpha)]^3}
58
59and
60
61.. math::
62
63    r(R_e,R_p,\alpha) = \left[ R_e^2 \sin^2 \alpha
64        + R_p^2 \cos^2 \alpha \right]^{1/2}
65
66
67$\alpha$ is the angle between the axis of the ellipsoid and $\vec q$,
68$V = (4/3)\pi R_pR_e^2$ is the volume of the ellipsoid , $R_p$ is the polar radius along the
69rotational axis of the ellipsoid, $R_e$ is the equatorial radius perpendicular
70to the rotational axis of the ellipsoid and $\Delta \rho$ (contrast) is the
71scattering length density difference, either $(sld\_core - sld\_shell)$ or $(sld\_shell - sld\_solvent)$.
72
73For randomly oriented particles:
74
75.. math::
76
77   F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}
78
79For oriented ellipsoids the *theta*, *phi* and *psi* orientation parameters will appear when fitting 2D data,
80see the :ref:`elliptical-cylinder` model for further information.
81
82References
83----------
84see for example:
85Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys., 1983, 79, 2461.
86Berr, S.  J. Phys. Chem., 1987, 91, 4760.
87
88Authorship and Verification
89----------------------------
90
91* **Author:** NIST IGOR/DANSE **Date:** pre 2010
92* **Last Modified by:** Richard Heenan (reparametrised model) **Date:** 2015
93* **Last Reviewed by:** Richard Heenan **Date:** October 6, 2016
94
95"""
96
97from numpy import inf, sin, cos, pi
98
99name = "core_shell_ellipsoid"
100title = "Form factor for an spheroid ellipsoid particle with a core shell structure."
101description = """
102        [core_shell_ellipsoid] Calculates the form factor for an spheroid
103        ellipsoid particle with a core_shell structure.
104        The form factor is averaged over all possible
105        orientations of the ellipsoid such that P(q)
106        = scale*<f^2>/Vol + bkg, where f is the
107        single particle scattering amplitude.
108        [Parameters]:
109        radius_equat_core = equatorial radius of core,
110        x_core = ratio of core polar/equatorial radii,
111        thick_shell = equatorial radius of outer surface,
112        x_polar_shell = ratio of polar shell thickness to equatorial shell thickness,
113        sld_core = SLD_core
114        sld_shell = SLD_shell
115        sld_solvent = SLD_solvent
116        background = Incoherent bkg
117        scale =scale
118        Note:It is the users' responsibility to ensure
119        that shell radii are larger than core radii.
120        oblate: polar radius < equatorial radius
121        prolate :  polar radius > equatorial radius - this new model will make this easier
122        and polydispersity integrals more logical (as previously the shell could disappear).
123    """
124category = "shape:ellipsoid"
125
126# pylint: disable=bad-whitespace, line-too-long
127#             ["name", "units", default, [lower, upper], "type", "description"],
128parameters = [
129    ["radius_equat_core","Ang",     20,   [0, inf],   "volume",      "Equatorial radius of core"],
130    ["x_core",        "None",       3,   [0, inf],    "volume",      "axial ratio of core, X = r_polar/r_equatorial"],
131    ["thick_shell",   "Ang",       30,   [0, inf],    "volume",      "thickness of shell at equator"],
132    ["x_polar_shell", "",           1,   [0, inf],    "volume",      "ratio of thickness of shell at pole to that at equator"],
133    ["sld_core",      "1e-6/Ang^2", 2,   [-inf, inf], "sld",         "Core scattering length density"],
134    ["sld_shell",     "1e-6/Ang^2", 1,   [-inf, inf], "sld",         "Shell scattering length density"],
135    ["sld_solvent",   "1e-6/Ang^2", 6.3, [-inf, inf], "sld",         "Solvent scattering length density"],
136    ["theta",         "degrees",    0,   [-360, 360], "orientation", "elipsoid axis to beam angle"],
137    ["phi",           "degrees",    0,   [-360, 360], "orientation", "rotation about beam"],
138    ]
139# pylint: enable=bad-whitespace, line-too-long
140
141source = ["lib/sas_3j1x_x.c", "lib/gauss76.c", "core_shell_ellipsoid.c"]
142
143def ER(radius_equat_core, x_core, thick_shell, x_polar_shell):
144    """
145        Returns the effective radius used in the S*P calculation
146    """
147    from .ellipsoid import ER as ellipsoid_ER
148    polar_outer = radius_equat_core*x_core + thick_shell*x_polar_shell
149    equat_outer = radius_equat_core + thick_shell
150    return ellipsoid_ER(polar_outer, equat_outer)
151
152
153demo = dict(scale=0.05, background=0.001,
154            radius_equat_core=20.0,
155            x_core=3.0,
156            thick_shell=30.0,
157            x_polar_shell=1.0,
158            sld_core=2.0,
159            sld_shell=1.0,
160            sld_solvent=6.3,
161            theta=0,
162            phi=0)
163
164q = 0.1
165# tests had in old coords theta=0, phi=0; new coords theta=90, phi=0
166qx = q*cos(pi/6.0)
167qy = q*sin(pi/6.0)
168# 11Jan2017 RKH sorted tests after redefinition of angles
169tests = [
170     # Accuracy tests based on content in test/utest_coreshellellipsoidXTmodel.py
171    [{'radius_equat_core': 200.0,
172      'x_core': 0.1,
173      'thick_shell': 50.0,
174      'x_polar_shell': 0.2,
175      'sld_core': 2.0,
176      'sld_shell': 1.0,
177      'sld_solvent': 6.3,
178      'background': 0.001,
179      'scale': 1.0,
180     }, 1.0, 0.00189402],
181
182    # Additional tests with larger range of parameters
183    [{'background': 0.01}, 0.1, 11.6915],
184
185    [{'radius_equat_core': 20.0,
186      'x_core': 200.0,
187      'thick_shell': 54.0,
188      'x_polar_shell': 3.0,
189      'sld_core': 20.0,
190      'sld_shell': 10.0,
191      'sld_solvent': 6.0,
192      'background': 0.0,
193      'scale': 1.0,
194     }, 0.01, 8688.53],
195
196   # 2D tests
197   [{'background': 0.001,
198     'theta': 90.0,
199     'phi': 0.0,
200     }, (0.4, 0.5), 0.00690673],
201
202   [{'radius_equat_core': 20.0,
203      'x_core': 200.0,
204      'thick_shell': 54.0,
205      'x_polar_shell': 3.0,
206      'sld_core': 20.0,
207      'sld_shell': 10.0,
208      'sld_solvent': 6.0,
209      'background': 0.01,
210      'scale': 0.01,
211      'theta': 90.0,
212      'phi': 0.0,
213     }, (qx, qy), 0.01000025],
214    ]
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