# source:sasmodels/sasmodels/models/core_shell_cylinder.py@0507e09

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
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1r"""
2Definition
3----------
4
5The output of the 2D scattering intensity function for oriented core-shell
6cylinders is given by (Kline, 2006 [#kline]_). The form factor is normalized
7by the particle volume. Note that in this model the shell envelops the entire
8core so that besides a "sleeve" around the core, the shell also provides two
9flat end caps of thickness = shell thickness. In other words the length of the
10total cyclinder is the length of the core cylinder plus twice the thickness of
11the shell. If no end caps are desired one should use the
12:ref:core-shell-bicelle and set the thickness of the end caps (in this case
13the "thick_face") to zero.
14
15.. math::
16
17    I(q,\alpha) = \frac{\text{scale}}{V_s} F^2(q,\alpha).sin(\alpha) + \text{background}
18
19where
20
21.. math::
22
23    F(q,\alpha) = &\ (\rho_c - \rho_s) V_c
24           \frac{\sin \left( q \tfrac12 L\cos\alpha \right)}
25                {q \tfrac12 L\cos\alpha}
26           \frac{2 J_1 \left( qR\sin\alpha \right)}
27                {qR\sin\alpha} \\
28         &\ + (\rho_s - \rho_\text{solv}) V_s
29           \frac{\sin \left( q \left(\tfrac12 L+T\right) \cos\alpha \right)}
30                {q \left(\tfrac12 L +T \right) \cos\alpha}
31           \frac{ 2 J_1 \left( q(R+T)\sin\alpha \right)}
32                {q(R+T)\sin\alpha}
33
34and
35
36.. math::
37
38    V_s = \pi (R + T)^2 (L + 2T)
39
40and $\alpha$ is the angle between the axis of the cylinder and $\vec q$,
41$V_s$ is the total volume (i.e. including both the core and the outer shell),
42$V_c$ is the volume of the core, $L$ is the length of the core,
43$R$ is the radius of the core, $T$ is the thickness of the shell, $\rho_c$
44is the scattering length density of the core, $\rho_s$ is the scattering
45length density of the shell, $\rho_\text{solv}$ is the scattering length
46density of the solvent, and *background* is the background level.  The outer
47radius of the shell is given by $R+T$ and the total length of the outer
48shell is given by $L+2T$. $J1$ is the first order Bessel function.
49
50.. _core-shell-cylinder-geometry:
51
52.. figure:: img/core_shell_cylinder_geometry.jpg
53
54    Core shell cylinder schematic.
55
56To provide easy access to the orientation of the core-shell cylinder, we
57define the axis of the cylinder using two angles $\theta$ and $\phi$.
58(see :ref:cylinder model <cylinder-angle-definition>)
59
60NB: The 2nd virial coefficient of the cylinder is calculated based on
61the radius and 2 length values, and used as the effective radius for
62$S(q)$ when $P(q) \cdot S(q)$ is applied.
63
64The $\theta$ and $\phi$ parameters are not used for the 1D output.
65
66Reference
67---------
68
69.. [#] see, for example, Ian Livsey  J. Chem. Soc., Faraday Trans. 2, 1987,83,
70   1445-1452
71.. [#kline] S R Kline, *J Appl. Cryst.*, 39 (2006) 895
72.. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659
73
74Source
75------
76
77core_shell_cylinder.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/core_shell_cylinder.py>_
78
79core_shell_cylinder.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/core_shell_cylinder.c>_
80
81Authorship and Verification
82----------------------------
83
84* **Author:** NIST IGOR/DANSE **Date:** pre 2010
86* **Last Reviewed by:** Richard Heenan **Date:** March 18, 2016
87* **Source added by :** Steve King **Date:** March 25, 2019
88"""
89
90import numpy as np
91from numpy import pi, inf, sin, cos
92
93name = "core_shell_cylinder"
94title = "Right circular cylinder with a core-shell scattering length density profile."
95description = """
96P(q,alpha)= scale/Vs*f(q)^(2) + background,
97      where: f(q)= 2(sld_core - solvant_sld)
98        * Vc*sin[qLcos(alpha/2)]
99        /[qLcos(alpha/2)]*J1(qRsin(alpha))
100        /[qRsin(alpha)]+2(sld_shell-sld_solvent)
101        *Vs*sin[q(L+T)cos(alpha/2)][[q(L+T)
102        *cos(alpha/2)]*J1(q(R+T)sin(alpha))
103        /q(R+T)sin(alpha)]
104
105    alpha:is the angle between the axis of
106        the cylinder and the q-vector
107    Vs: the volume of the outer shell
108    Vc: the volume of the core
109    L: the length of the core
110        sld_shell: the scattering length density of the shell
111    sld_solvent: the scattering length density of the solvent
112    background: the background
113    T: the thickness
114        R+T: is the outer radius
115     L+2T: The total length of the outershell
116    J1: the first order Bessel function
117     theta: axis_theta of the cylinder
118     phi: the axis_phi of the cylinder
119"""
120category = "shape:cylinder"
121
122#             ["name", "units", default, [lower, upper], "type", "description"],
123parameters = [["sld_core", "1e-6/Ang^2", 4, [-inf, inf], "sld",
124               "Cylinder core scattering length density"],
125              ["sld_shell", "1e-6/Ang^2", 4, [-inf, inf], "sld",
126               "Cylinder shell scattering length density"],
127              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
128               "Solvent scattering length density"],
129              ["radius", "Ang", 20, [0, inf], "volume",
131              ["thickness", "Ang", 20, [0, inf], "volume",
132               "Cylinder shell thickness"],
133              ["length", "Ang", 400, [0, inf], "volume",
134               "Cylinder length"],
135              ["theta", "degrees", 60, [-360, 360], "orientation",
136               "cylinder axis to beam angle"],
137              ["phi", "degrees", 60, [-360, 360], "orientation",
139             ]
140
141source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "core_shell_cylinder.c"]
142have_Fq = True
144    "excluded volume", "equivalent volume sphere", "outer radius", "half outer length",
145    "half min outer dimension", "half max outer dimension", "half outer diagonal",
146    ]
147
148def random():
149    """Return a random parameter set for the model."""
151    # Use a distribution with a preference for thin shell or thin core
152    # Avoid core,shell radii < 1
155    length = np.random.uniform(1, 4.7)
156    pars = dict(
158        thickness=thickness,
159        length=length,
160    )
161    return pars
162
163demo = dict(scale=1, background=0,
164            sld_core=6, sld_shell=8, sld_solvent=1,
166            theta=30, phi=15,
168            length_pd=.2, length_pd_n=10,
169            thickness_pd=.2, thickness_pd_n=10,
170            theta_pd=15, theta_pd_n=45,
171            phi_pd=15, phi_pd_n=1)
172q = 0.1
173# april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct!
174qx = q*cos(pi/6.0)
175qy = q*sin(pi/6.0)
176tests = [
177    [{}, 0.075, 10.8552692237],
178    [{}, (qx, qy), 0.444618752741],
179]
180del qx, qy  # not necessary to delete, but cleaner
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