source: sasmodels/sasmodels/models/core_shell_bicelle.py @ 99658f6

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 99658f6 was 99658f6, checked in by grethevj, 10 months ago

updated ER functions including cylinder excluded volume, to match 4.x

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Line 
1r"""
2Definition
3----------
4
5This model provides the form factor for a circular cylinder with a
6core-shell scattering length density profile. Thus this is a variation
7of a core-shell cylinder or disc where the shell on the walls and ends
8may be of different thicknesses and scattering length densities. The form
9factor is normalized by the 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 cylindrical symmetry 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 cylinder 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, to give:
39
40.. math::
41
42    I(Q,\alpha) = \frac{\text{scale}}{V_t} \cdot
43        F(Q,\alpha)^2 \cdot sin(\alpha) + \text{background}
44
45where
46
47.. math::
48    :nowrap:
49
50    \begin{align*}
51    F(Q,\alpha) = &\bigg[
52    (\rho_c - \rho_f) V_c
53     \frac{2J_1(QRsin \alpha)}{QRsin\alpha}
54     \frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\
55    &+(\rho_f - \rho_r) V_{c+f}
56     \frac{2J_1(QRsin\alpha)}{QRsin\alpha}
57     \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha} \\
58    &+(\rho_r - \rho_s) V_t
59     \frac{2J_1(Q(R+t_r)sin\alpha)}{Q(R+t_r)sin\alpha}
60     \frac{sin(Q(L/2+t_f)cos\alpha)}{Q(L/2+t_f)cos\alpha}
61    \bigg]
62    \end{align*}
63
64where $V_t$ is the total volume of the bicelle, $V_c$ the volume of the core,
65$V_{c+f}$ the volume of the core plus the volume of the faces, $R$ is the radius
66of the core, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$
67the thickness of the rim and $J_1$ the usual first order bessel function.
68
69The output of the 1D scattering intensity function for randomly oriented
70cylinders is then given by integrating over all possible $\theta$ and $\phi$.
71
72For oriented bicelles the *theta*, and *phi* orientation parameters will appear
73when fitting 2D data, see the :ref:`cylinder` model for further information.
74Our implementation of the scattering kernel and the 1D scattering intensity
75use the c-library from NIST.
76
77.. figure:: img/cylinder_angle_definition.png
78
79    Definition of the angles for the oriented core shell bicelle model,
80    note that the cylinder axis of the bicelle starts along the beam direction
81    when $\theta  = \phi = 0$.
82
83
84References
85----------
86
87.. [#] D Singh (2009). *Small angle scattering studies of self assembly in
88   lipid mixtures*, John's Hopkins University Thesis (2009) 223-225. `Available
89   from Proquest <http://search.proquest.com/docview/304915826?accountid
90   =26379>`_
91   
92   L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949).
93
94Authorship and Verification
95----------------------------
96
97* **Author:** NIST IGOR/DANSE **Date:** pre 2010
98* **Last Modified by:** Paul Butler **Date:** September 30, 2016
99* **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017
100"""
101
102import numpy as np
103from numpy import inf, sin, cos, pi
104
105name = "core_shell_bicelle"
106title = "Circular cylinder with a core-shell scattering length density profile.."
107description = """
108    P(q,alpha)= (scale/Vs)*f(q)^(2) + bkg,  where:
109    f(q)= Vt(sld_rim - sld_solvent)* sin[qLt.cos(alpha)/2]
110    /[qLt.cos(alpha)/2]*J1(qRout.sin(alpha))
111    /[qRout.sin(alpha)]+
112    (sld_core-sld_face)*Vc*sin[qLcos(alpha)/2][[qL
113    *cos(alpha)/2]*J1(qRc.sin(alpha))
114    /qRc.sin(alpha)]+
115    (sld_face-sld_rim)*(Vc+Vf)*sin[q(L+2.thick_face).
116    cos(alpha)/2][[q(L+2.thick_face)*cos(alpha)/2]*
117    J1(qRc.sin(alpha))/qRc.sin(alpha)]
118
119    alpha:is the angle between the axis of
120    the cylinder and the q-vector
121    Vt = pi.(Rc + thick_rim)^2.Lt : total volume
122    Vc = pi.Rc^2.L :the volume of the core
123    Vf = 2.pi.Rc^2.thick_face
124    Rc = radius: is the core radius
125    L: the length of the core
126    Lt = L + 2.thick_face: total length
127    Rout = radius + thick_rim
128    sld_core, sld_rim, sld_face:scattering length
129    densities within the particle
130    sld_solvent: the scattering length density
131    of the solvent
132    bkg: the background
133    J1: the first order Bessel function
134    theta: axis_theta of the cylinder
135    phi: the axis_phi of the cylinder...
136        """
137category = "shape:cylinder"
138
139# pylint: disable=bad-whitespace, line-too-long
140#             ["name", "units", default, [lower, upper], "type", "description"],
141parameters = [
142    ["radius",         "Ang",       80, [0, inf],    "volume",      "Cylinder core radius"],
143    ["thick_rim",  "Ang",       10, [0, inf],    "volume",      "Rim shell thickness"],
144    ["thick_face", "Ang",       10, [0, inf],    "volume",      "Cylinder face thickness"],
145    ["length",         "Ang",      50, [0, inf],    "volume",      "Cylinder length"],
146    ["sld_core",       "1e-6/Ang^2", 1, [-inf, inf], "sld",         "Cylinder core scattering length density"],
147    ["sld_face",       "1e-6/Ang^2", 4, [-inf, inf], "sld",         "Cylinder face scattering length density"],
148    ["sld_rim",        "1e-6/Ang^2", 4, [-inf, inf], "sld",         "Cylinder rim scattering length density"],
149    ["sld_solvent",    "1e-6/Ang^2", 1, [-inf, inf], "sld",         "Solvent scattering length density"],
150    ["theta",          "degrees",   90, [-360, 360], "orientation", "cylinder axis to beam angle"],
151    ["phi",            "degrees",    0, [-360, 360], "orientation", "rotation about beam"]
152    ]
153
154# pylint: enable=bad-whitespace, line-too-long
155
156source = ["lib/sas_Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c",
157          "core_shell_bicelle.c"]
158have_Fq = True
159effective_radius_type = [
160    "excluded volume","equivalent volume sphere", "outer rim radius",
161    "half outer thickness", "half diagonal",
162    ]
163
164def random():
165    pars = dict(
166        radius=10**np.random.uniform(1.3, 3),
167        length=10**np.random.uniform(1.3, 4),
168        thick_rim=10**np.random.uniform(0, 1.7),
169        thick_face=10**np.random.uniform(0, 1.7),
170    )
171    return pars
172
173demo = dict(scale=1, background=0,
174            radius=20.0,
175            thick_rim=10.0,
176            thick_face=10.0,
177            length=400.0,
178            sld_core=1.0,
179            sld_face=4.0,
180            sld_rim=4.0,
181            sld_solvent=1.0,
182            theta=90,
183            phi=0)
184q = 0.1
185# april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct!
186qx = q*cos(pi/6.0)
187qy = q*sin(pi/6.0)
188tests = [
189    [{}, 0.05, 7.4883545957],
190    [{'theta':80., 'phi':10.}, (qx, qy), 2.81048892474]
191]
192del qx, qy  # not necessary to delete, but cleaner
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