source: sasmodels/sasmodels/models/core_shell_bicelle.py @ 30b60d2

core_shell_microgelscostrafo411magnetic_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 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 \frac{2J_1(QRsin \alpha)}{QRsin\alpha}\frac{sin(QLcos\alpha/2)}{Q(L/2)cos\alpha} \\
53    &+(\rho_f - \rho_r) V_{c+f} \frac{2J_1(QRsin\alpha)}{QRsin\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 $V_t$ is the total volume of the bicelle, $V_c$ the volume of the core,
59$V_{c+f}$ the volume of the core plus the volume of the faces, $R$ is the radius
60of the core, $L$ the length of the core, $t_f$ the thickness of the face, $t_r$
61the thickness of the rim and $J_1$ the usual first order bessel function.
62
63The output of the 1D scattering intensity function for randomly oriented
64cylinders is then given by integrating over all possible $\theta$ and $\phi$.
65
66For oriented bicelles the *theta*, and *phi* orientation parameters will appear when fitting 2D data,
67see the :ref:`cylinder` model for further information.
68Our implementation of the scattering kernel and the 1D scattering intensity
69use the c-library from NIST.
70
71.. figure:: img/cylinder_angle_definition.png
72
73    Definition of the angles for the oriented core shell bicelle model,
74    note that the cylinder axis of the bicelle starts along the beam direction
75    when $\theta  = \phi = 0$.
76
77
78References
79----------
80
81.. [#] D Singh (2009). *Small angle scattering studies of self assembly in
82   lipid mixtures*, John's Hopkins University Thesis (2009) 223-225. `Available
83   from Proquest <http://search.proquest.com/docview/304915826?accountid
84   =26379>`_
85
86Authorship and Verification
87----------------------------
88
89* **Author:** NIST IGOR/DANSE **Date:** pre 2010
90* **Last Modified by:** Paul Butler **Date:** September 30, 2016
91* **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017
92"""
93
94from numpy import inf, sin, cos, pi
95
96name = "core_shell_bicelle"
97title = "Circular cylinder with a core-shell scattering length density profile.."
98description = """
99    P(q,alpha)= (scale/Vs)*f(q)^(2) + bkg,  where:
100    f(q)= Vt(sld_rim - sld_solvent)* sin[qLt.cos(alpha)/2]
101    /[qLt.cos(alpha)/2]*J1(qRout.sin(alpha))
102    /[qRout.sin(alpha)]+
103    (sld_core-sld_face)*Vc*sin[qLcos(alpha)/2][[qL
104    *cos(alpha)/2]*J1(qRc.sin(alpha))
105    /qRc.sin(alpha)]+
106    (sld_face-sld_rim)*(Vc+Vf)*sin[q(L+2.thick_face).
107    cos(alpha)/2][[q(L+2.thick_face)*cos(alpha)/2]*
108    J1(qRc.sin(alpha))/qRc.sin(alpha)]
109
110    alpha:is the angle between the axis of
111    the cylinder and the q-vector
112    Vt = pi.(Rc + thick_rim)^2.Lt : total volume
113    Vc = pi.Rc^2.L :the volume of the core
114    Vf = 2.pi.Rc^2.thick_face
115    Rc = radius: is the core radius
116    L: the length of the core
117    Lt = L + 2.thick_face: total length
118    Rout = radius + thick_rim
119    sld_core, sld_rim, sld_face:scattering length
120    densities within the particle
121    sld_solvent: the scattering length density
122    of the solvent
123    bkg: the background
124    J1: the first order Bessel function
125    theta: axis_theta of the cylinder
126    phi: the axis_phi of the cylinder...
127        """
128category = "shape:cylinder"
129
130# pylint: disable=bad-whitespace, line-too-long
131#             ["name", "units", default, [lower, upper], "type", "description"],
132parameters = [
133    ["radius",         "Ang",       80, [0, inf],    "volume",      "Cylinder core radius"],
134    ["thick_rim",  "Ang",       10, [0, inf],    "volume",      "Rim shell thickness"],
135    ["thick_face", "Ang",       10, [0, inf],    "volume",      "Cylinder face thickness"],
136    ["length",         "Ang",      50, [0, inf],    "volume",      "Cylinder length"],
137    ["sld_core",       "1e-6/Ang^2", 1, [-inf, inf], "sld",         "Cylinder core scattering length density"],
138    ["sld_face",       "1e-6/Ang^2", 4, [-inf, inf], "sld",         "Cylinder face scattering length density"],
139    ["sld_rim",        "1e-6/Ang^2", 4, [-inf, inf], "sld",         "Cylinder rim scattering length density"],
140    ["sld_solvent",    "1e-6/Ang^2", 1, [-inf, inf], "sld",         "Solvent scattering length density"],
141    ["theta",          "degrees",   90, [-360, 360], "orientation", "cylinder axis to beam angle"],
142    ["phi",            "degrees",    0, [-360, 360], "orientation", "rotation about beam"]
143    ]
144
145# pylint: enable=bad-whitespace, line-too-long
146
147source = ["lib/sas_Si.c", "lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c",
148          "core_shell_bicelle.c"]
149
150def random():
151    import numpy as np
152    pars = dict(
153        radius=10**np.random.uniform(1.3, 3),
154        length=10**np.random.uniform(1.3, 4),
155        thick_rim=10**np.random.uniform(0, 1.7),
156        thick_face=10**np.random.uniform(0, 1.7),
157    )
158    return pars
159
160demo = dict(scale=1, background=0,
161            radius=20.0,
162            thick_rim=10.0,
163            thick_face=10.0,
164            length=400.0,
165            sld_core=1.0,
166            sld_face=4.0,
167            sld_rim=4.0,
168            sld_solvent=1.0,
169            theta=90,
170            phi=0)
171q = 0.1
172# april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct!
173qx = q*cos(pi/6.0)
174qy = q*sin(pi/6.0)
175tests = [[{}, 0.05, 7.4883545957],
176        [{'theta':80., 'phi':10.}, (qx, qy), 2.81048892474 ]
177        ]
178del qx, qy  # not necessary to delete, but cleaner
179
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