source: sasmodels/sasmodels/models/capped_cylinder.py @ fcb33e4

core_shell_microgelscostrafo411magnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since fcb33e4 was fcb33e4, checked in by richardh, 7 years ago

new model core_shell_bicelle_elliptical, not tested for 2d, docu changes for other cylinder models

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Line 
1r"""
2Definitions
3-----------
4
5Calculates the scattering from a cylinder with spherical section end-caps.
6Like :ref:`barbell`, this is a sphereocylinder with end caps that have a
7radius larger than that of the cylinder, but with the center of the end cap
8radius lying within the cylinder. This model simply becomes a convex
9lens when the length of the cylinder $L=0$. See the diagram for the details
10of the geometry and restrictions on parameter values.
11
12.. figure:: img/capped_cylinder_geometry.jpg
13
14    Capped cylinder geometry, where $r$ is *radius*, $R$ is *bell_radius* and
15    $L$ is *length*. Since the end cap radius $R \geq r$ and by definition
16    for this geometry $h < 0$, $h$ is then defined by $r$ and $R$ as
17    $h = - \sqrt{R^2 - r^2}$
18
19The scattered intensity $I(q)$ is calculated as
20
21.. math::
22
23    I(q) = \frac{\Delta \rho^2}{V} \left<A^2(q,\alpha).sin(\alpha)\right>
24
25where the amplitude $A(q,\alpha)$ with the rod axis at angle $\alpha$ to $q$ is given as
26
27.. math::
28
29    A(q) =&\ \pi r^2L
30        \frac{\sin\left(\tfrac12 qL\cos\alpha\right)}
31            {\tfrac12 qL\cos\alpha}
32        \frac{2 J_1(qr\sin\alpha)}{qr\sin\alpha} \\
33        &\ + 4 \pi R^3 \int_{-h/R}^1 dt
34        \cos\left[ q\cos\alpha
35            \left(Rt + h + {\tfrac12} L\right)\right]
36        \times (1-t^2)
37        \frac{J_1\left[qR\sin\alpha \left(1-t^2\right)^{1/2}\right]}
38             {qR\sin\alpha \left(1-t^2\right)^{1/2}}
39
40The $\left<\ldots\right>$ brackets denote an average of the structure over
41all orientations. $\left< A^2(q)\right>$ is then the form factor, $P(q)$.
42The scale factor is equivalent to the volume fraction of cylinders, each of
43volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length
44densities of the cylinder and the surrounding solvent.
45
46The volume of the capped cylinder is (with $h$ as a positive value here)
47
48.. math::
49
50    V = \pi r_c^2 L + \tfrac{2\pi}{3}(R-h)^2(2R + h)
51
52
53and its radius of gyration is
54
55.. math::
56
57    R_g^2 =&\ \left[ \tfrac{12}{5}R^5
58        + R^4\left(6h+\tfrac32 L\right)
59        + R^2\left(4h^2 + L^2 + 4Lh\right)
60        + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\
61        &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3
62        + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right]
63        \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1}
64
65
66.. note::
67
68    The requirement that $R \geq r$ is not enforced in the model!
69    It is up to you to restrict this during analysis.
70
71The 2D scattering intensity is calculated similar to the 2D cylinder model.
72
73.. figure:: img/cylinder_angle_definition.jpg
74
75    Definition of the angles for oriented 2D cylinders.
76
77
78References
79----------
80
81.. [#] H Kaya, *J. Appl. Cryst.*, 37 (2004) 223-230
82.. [#] H Kaya and N-R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda
83   and errata)
84
85Authorship and Verification
86----------------------------
87
88* **Author:** NIST IGOR/DANSE **Date:** pre 2010
89* **Last Modified by:** Paul Butler **Date:** September 30, 2016
90* **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017
91
92"""
93from numpy import inf
94
95name = "capped_cylinder"
96title = "Right circular cylinder with spherical end caps and uniform SLD"
97description = """That is, a sphereocylinder
98    with end caps that have a radius larger than
99    that of the cylinder and the center of the
100    end cap radius lies within the cylinder.
101    Note: As the length of cylinder -->0,
102    it becomes a Convex Lens.
103    It must be that radius <(=) radius_cap.
104    [Parameters];
105    scale: volume fraction of spheres,
106    background:incoherent background,
107    radius: radius of the cylinder,
108    length: length of the cylinder,
109    radius_cap: radius of the semi-spherical cap,
110    sld: SLD of the capped cylinder,
111    sld_solvent: SLD of the solvent.
112"""
113category = "shape:cylinder"
114# pylint: disable=bad-whitespace, line-too-long
115#             ["name", "units", default, [lower, upper], "type", "description"],
116parameters = [["sld",         "1e-6/Ang^2", 4, [-inf, inf], "sld",    "Cylinder scattering length density"],
117              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",    "Solvent scattering length density"],
118              ["radius",      "Ang",       20, [0, inf],    "volume", "Cylinder radius"],
119
120              # TODO: use an expression for cap radius with fixed bounds.
121              # The current form requires cap radius R bigger than cylinder radius r.
122              # Could instead use R/r in [1,inf], r/R in [0,1], or the angle between
123              # cylinder and cap in [0,90].  The problem is similar for the barbell
124              # model.  Propose r/R in [0,1] in both cases, with the model specifying
125              # cylinder radius in the capped cylinder model and sphere radius in the
126              # barbell model.  This leads to the natural value of zero for no cap
127              # in the capped cylinder, and zero for no bar in the barbell model.  In
128              # both models, one would be a pill.
129              ["radius_cap", "Ang",     20, [0, inf],    "volume", "Cap radius"],
130              ["length",     "Ang",    400, [0, inf],    "volume", "Cylinder length"],
131              ["theta",      "degrees", 60, [-inf, inf], "orientation", "inclination angle"],
132              ["phi",        "degrees", 60, [-inf, inf], "orientation", "deflection angle"],
133             ]
134# pylint: enable=bad-whitespace, line-too-long
135
136source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "capped_cylinder.c"]
137
138demo = dict(scale=1, background=0,
139            sld=6, sld_solvent=1,
140            radius=260, radius_cap=290, length=290,
141            theta=30, phi=15,
142            radius_pd=.2, radius_pd_n=1,
143            radius_cap_pd=.2, radius_cap_pd_n=1,
144            length_pd=.2, length_pd_n=10,
145            theta_pd=15, theta_pd_n=45,
146            phi_pd=15, phi_pd_n=1)
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