# source:sasmodels/sasmodels/models/core_shell_bicelle_elliptical.py@b297ba9

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
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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
10particle volume.
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 $\psi$
39is the angle for 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 \cdot sin(\alpha) + \text{background}
45
46where a numerical integration of $F(Q,\alpha, \psi)^2 \cdot sin(\alpha)$
47is carried out over \alpha and \psi for:
48
49.. math::
50    :nowrap:
51
52    \begin{align*}
53    F(Q,\alpha,\psi) = &\bigg[
54    (\rho_c - \rho_f) V_c
55     \frac{2J_1(QR'sin \alpha)}{QR'sin\alpha}
56     \frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\
57    &+(\rho_f - \rho_r) V_{c+f}
58     \frac{2J_1(QR'sin\alpha)}{QR'sin\alpha}
59     \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} \\
60    &+(\rho_r - \rho_s) V_t
61     \frac{2J_1(Q(R'+t_r)sin\alpha)}{Q(R'+t_r)sin\alpha}
62     \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha}
63    \bigg]
64    \end{align*}
65
66where
67
68.. math::
69
70    R'=\frac{R}{\sqrt{2}}\sqrt{(1+X_{core}^{2}) + (1-X_{core}^{2})cos(\psi)}
71
72
73and $V_t = \pi.(R+t_r)(Xcore.R+t_r)^2.(L+2.t_f)$ is the total volume of
74the bicelle, $V_c = \pi.Xcore.R^2.L$ the volume of the core,
75$V_{c+f} = \pi.Xcore.R^2.(L+2.t_f)$ the volume of the core plus the volume
76of the faces, $R$ is the radius of the core, $Xcore$ is the axial ratio of
77the core, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$
78the thickness of the rim and $J_1$ the usual first order bessel function.
79The core has radii $R$ and $Xcore.R$ so is circular, as for the
80core_shell_bicelle model, for $Xcore$ =1. Note that you may need to
81limit the range of $Xcore$, especially if using the Monte-Carlo algorithm,
82as setting radius to $R/Xcore$ and axial ratio to $1/Xcore$ gives an
83equivalent solution!
84
85The output of the 1D scattering intensity function for randomly oriented
86bicelles is then given by integrating over all possible $\alpha$ and $\psi$.
87
88For oriented bicelles the *theta*, *phi* and *psi* orientation parameters will
89appear when fitting 2D data, see the :ref:elliptical-cylinder model
90for further information.
91
92.. figure:: img/elliptical_cylinder_angle_definition.png
93
94    Definition of the angles for the oriented core_shell_bicelle_elliptical particles.
95
96Model verified using Monte Carlo simulation for 1D and 2D scattering.
97
98References
99----------
100
101.. [#]
102L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949).
103
104Authorship and Verification
105----------------------------
106
107* **Author:** Richard Heenan **Date:** December 14, 2016
109* **Last Reviewed by:**  Paul Kienzle **Date:** Feb 28, 2018
110"""
111
112import numpy as np
113from numpy import inf, sin, cos, pi
114
115name = "core_shell_bicelle_elliptical"
116title = "Elliptical cylinder with a core-shell scattering length density profile.."
117description = """
118    core_shell_bicelle_elliptical
119    Elliptical cylinder core, optional shell on the two flat faces, and shell of
120    uniform thickness on its rim (extending around the end faces).
121    Please see full documentation for equations and further details.
122    Involves a double numerical integral around the ellipsoid diameter
123    and the angle of the cylinder axis to Q.
124    Compare also the core_shell_bicelle and elliptical_cylinder models.
125      """
126category = "shape:cylinder"
127
129#             ["name", "units", default, [lower, upper], "type", "description"],
130parameters = [
132    ["x_core",        "None",       3,  [0, inf],    "volume",      "Axial ratio of core, X = r_major/r_minor"],
133    ["thick_rim",  "Ang",            8, [0, inf],    "volume",      "Rim shell thickness"],
134    ["thick_face", "Ang",           14, [0, inf],    "volume",      "Cylinder face thickness"],
135    ["length",         "Ang",       50, [0, inf],    "volume",      "Cylinder length"],
136    ["sld_core",       "1e-6/Ang^2", 4, [-inf, inf], "sld",         "Cylinder core scattering length density"],
137    ["sld_face",       "1e-6/Ang^2", 7, [-inf, inf], "sld",         "Cylinder face scattering length density"],
138    ["sld_rim",        "1e-6/Ang^2", 1, [-inf, inf], "sld",         "Cylinder rim scattering length density"],
139    ["sld_solvent",    "1e-6/Ang^2", 6, [-inf, inf], "sld",         "Solvent scattering length density"],
140    ["theta",       "degrees",    90.0, [-360, 360], "orientation", "Cylinder axis to beam angle"],
141    ["phi",         "degrees",    0,    [-360, 360], "orientation", "Rotation about beam"],
142    ["psi",         "degrees",    0,    [-360, 360], "orientation", "Rotation about cylinder axis"]
143    ]
144
146
147source = ["lib/sas_Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c",
148          "core_shell_bicelle_elliptical.c"]
149have_Fq = True
151    "equivalent cylinder excluded volume", "equivalent volume sphere",
153    "outer max radius", "half outer thickness", "half diagonal",
154    ]
155
156def random():
157    """Return a random parameter set for the model."""
158    outer_major = 10**np.random.uniform(1, 4.7)
159    outer_minor = 10**np.random.uniform(1, 4.7)
160    # Use a distribution with a preference for thin shell or thin core,
161    # limited by the minimum radius. Avoid core,shell radii < 1
163    thick_rim = np.random.beta(0.5, 0.5)*(min_radius-2) + 1
164    radius_major = outer_major - thick_rim
165    radius_minor = outer_minor - thick_rim
168    outer_length = 10**np.random.uniform(1, 4.7)
169    # Caps should be a small percentage of the total length, but at least one
170    # angstrom long.  Since outer length >= 10, the following will suffice
171    thick_face = 10**np.random.uniform(-np.log10(outer_length), -1)*outer_length
172    length = outer_length - thick_face
173    pars = dict(
175        x_core=x_core,
176        thick_rim=thick_rim,
177        thick_face=thick_face,
178        length=length
179    )
180    return pars
181
182
183q = 0.1
184# april 6 2017, rkh added a 2d unit test, NOT READY YET pull #890 branch assume correct!
185qx = q*cos(pi/6.0)
186qy = q*sin(pi/6.0)
187
188tests = [
190    #  'thick_rim': 8.0, 'thick_face': 14.0, 'length': 50.0}, 'ER', 1],