source: sasmodels/sasmodels/models/core_shell_bicelle_elliptical.py @ 8e68ea0

core_shell_microgelscostrafo411magnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 8e68ea0 was 8e68ea0, checked in by wojciech, 6 years ago

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1r"""
2Definition
3----------
4
5This model provides the form factor for an elliptical cylinder with a
6core-shell scattering length density profile. Thus this is a variation
7of the core-shell bicelle model, but with an elliptical cylinder for the core.
8Outer shells on the rims and flat ends may be of different thicknesses and
9scattering length densities. The form factor is normalized by the total particle volume.
10
11
12.. figure:: img/core_shell_bicelle_geometry.png
13
14    Schematic cross-section of bicelle. Note however that the model here
15    calculates for rectangular, not curved, rims as shown below.
16
17.. figure:: img/core_shell_bicelle_parameters.png
18
19   Cross section of model used here. Users will have
20   to decide how to distribute "heads" and "tails" between the rim, face
21   and core regions in order to estimate appropriate starting parameters.
22
23Given the scattering length densities (sld) $\rho_c$, the core sld, $\rho_f$,
24the face sld, $\rho_r$, the rim sld and $\rho_s$ the solvent sld, the
25scattering length density variation along the bicelle axis is:
26
27.. math::
28
29    \rho(r) =
30      \begin{cases}
31      &\rho_c \text{ for } 0 \lt r \lt R; -L \lt z\lt L \\[1.5ex]
32      &\rho_f \text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L;
33      L \lt z\lt (L+2t) \\[1.5ex]
34      &\rho_r\text{ for } 0 \lt r \lt R; -(L+2t) \lt z\lt -L; L \lt z\lt (L+2t)
35      \end{cases}
36
37The form factor for the bicelle is calculated in cylindrical coordinates, where
38$\alpha$ is the angle between the $Q$ vector and the cylinder axis, and $\psi$ is the angle
39for the ellipsoidal cross section core, to give:
40
41.. math::
42
43    I(Q,\alpha,\psi) = \frac{\text{scale}}{V_t} \cdot
44        F(Q,\alpha, \psi)^2.sin(\alpha) + \text{background}
45
46where a numerical integration of $F(Q,\alpha, \psi)^2.sin(\alpha)$ is carried out over \alpha and \psi for:
47
48.. math::
49
50        \begin{align}   
51    F(Q,\alpha,\psi) = &\bigg[
52    (\rho_c - \rho_f) V_c \frac{2J_1(QR'sin \alpha)}{QR'sin\alpha}\frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\
53    &+(\rho_f - \rho_r) V_{c+f} \frac{2J_1(QR'sin\alpha)}{QR'sin\alpha}\frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} \\
54    &+(\rho_r - \rho_s) V_t \frac{2J_1(Q(R'+t_r)sin\alpha)}{Q(R'+t_r)sin\alpha}\frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha}
55    \bigg]
56    \end{align}
57
58where
59
60.. math::
61
62    R'=\frac{R}{\sqrt{2}}\sqrt{(1+X_{core}^{2}) + (1-X_{core}^{2})cos(\psi)}
63   
64   
65and $V_t = \pi.(R+t_r)(Xcore.R+t_r)^2.(L+2.t_f)$ is the total volume of the bicelle,
66$V_c = \pi.Xcore.R^2.L$ the volume of the core, $V_{c+f} = \pi.Xcore.R^2.(L+2.t_f)$
67the volume of the core plus the volume of the faces, $R$ is the radius
68of the core, $Xcore$ is the axial ratio of the core, $L$ the length of the core,
69$t_f$ the thickness of the face, $t_r$ the thickness of the rim and $J_1$ the usual
70first order bessel function. The core has radii $R$ and $Xcore.R$ so is circular,
71as for the core_shell_bicelle model, for $Xcore$ =1. Note that you may need to
72limit the range of $Xcore$, especially if using the Monte-Carlo algorithm, as
73setting radius to $R/Xcore$ and axial ratio to $1/Xcore$ gives an equivalent solution!
74
75The output of the 1D scattering intensity function for randomly oriented
76bicelles is then given by integrating over all possible $\alpha$ and $\psi$.
77
78For oriented bicellles the *theta*, *phi* and *psi* orientation parameters only appear when fitting 2D data,
79see the :ref:`elliptical-cylinder` model for further information.
80
81
82.. figure:: img/elliptical_cylinder_angle_definition.jpg
83
84    Definition of the angles for the oriented core_shell_bicelle_elliptical model.
85
86
87References
88----------
89
90.. [#]
91
92Authorship and Verification
93----------------------------
94
95* **Author:** Richard Heenan **Date:** December 14, 2016
96* **Last Modified by:**  Richard Heenan **Date:** December 14, 2016
97* **Last Reviewed by:**  Richard Heenan BEWARE 2d data yet to be checked **Date:** December 14, 2016
98"""
99
100from numpy import inf, sin, cos
101
102name = "core_shell_bicelle_elliptical"
103title = "Elliptical cylinder with a core-shell scattering length density profile.."
104description = """
105    core_shell_bicelle_elliptical
106    Elliptical cylinder core, optional shell on the two flat faces, and shell of
107    uniform thickness on its rim (extending around the end faces).   
108    Please see full documentation for equations and further details.
109    Involves a double numerical integral around the ellipsoid diameter
110    and the angle of the cylinder axis to Q.
111    Compare also the core_shell_bicelle and elliptical_cylinder models.
112      """
113category = "shape:cylinder"
114
115# pylint: disable=bad-whitespace, line-too-long
116#             ["name", "units", default, [lower, upper], "type", "description"],
117parameters = [
118    ["radius",         "Ang",       30, [0, inf],    "volume",      "Cylinder core radius"],
119    ["x_core",        "None",       3,  [0, inf],    "volume",      "axial ratio of core, X = r_polar/r_equatorial"],
120    ["thick_rim",  "Ang",        8, [0, inf],    "volume",      "Rim shell thickness"],
121    ["thick_face", "Ang",       14, [0, inf],    "volume",      "Cylinder face thickness"],
122    ["length",         "Ang",      50, [0, inf],    "volume",      "Cylinder length"],
123    ["sld_core",       "1e-6/Ang^2", 4, [-inf, inf], "sld",         "Cylinder core scattering length density"],
124    ["sld_face",       "1e-6/Ang^2", 7, [-inf, inf], "sld",         "Cylinder face scattering length density"],
125    ["sld_rim",        "1e-6/Ang^2", 1, [-inf, inf], "sld",         "Cylinder rim scattering length density"],
126    ["sld_solvent",    "1e-6/Ang^2", 6, [-inf, inf], "sld",         "Solvent scattering length density"],
127    ["theta",          "degrees",   90, [-360, 360], "orientation", "In plane angle"],
128    ["phi",            "degrees",    0, [-360, 360], "orientation", "Out of plane angle"],
129    ["psi",            "degrees",    0, [-360, 360], "orientation", "Major axis angle relative to Q"],
130    ]
131
132# pylint: enable=bad-whitespace, line-too-long
133
134source = ["lib/sas_Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c",
135          "core_shell_bicelle_elliptical.c"]
136
137demo = dict(scale=1, background=0,
138            radius=30.0,
139            x_core=3.0,
140            thick_rim=8.0,
141            thick_face=14.0,
142            length=50.0,
143            sld_core=4.0,
144            sld_face=7.0,
145            sld_rim=1.0,
146            sld_solvent=6.0,
147            theta=90,
148            phi=0,
149            psi=0)
150
151qx, qy = 0.4 * cos(90), 0.5 * sin(0)
152
153tests = [
154    [{'radius': 30.0, 'x_core': 3.0, 'thick_rim':8.0, 'thick_face':14.0, 'length':50.0}, 'ER', 1],
155    [{'radius': 30.0, 'x_core': 3.0, 'thick_rim':8.0, 'thick_face':14.0, 'length':50.0}, 'VR', 1],
156
157    [{'radius': 30.0, 'x_core': 3.0, 'thick_rim':8.0, 'thick_face':14.0, 'length':50.0,
158    'sld_core':4.0, 'sld_face':7.0, 'sld_rim':1.0, 'sld_solvent':6.0, 'background':0.0},
159    0.015, 286.540286],
160]
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