source: sasmodels/sasmodels/models/capped_cylinder.py @ 0d6e865

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Last change on this file since 0d6e865 was 0d6e865, checked in by dirk, 7 years ago

Rewriting selected models in spherical coordinates. This fixes ticket #492 for these 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)\right>
24
25where the amplitude $A(q)$ is given as
26
27.. math::
28
29    A(q) =&\ \pi r^2L
30        \frac{\sin\left(\tfrac12 qL\cos\theta\right)}
31            {\tfrac12 qL\cos\theta}
32        \frac{2 J_1(qr\sin\theta)}{qr\sin\theta} \\
33        &\ + 4 \pi R^3 \int_{-h/R}^1 dt
34        \cos\left[ q\cos\theta
35            \left(Rt + h + {\tfrac12} L\right)\right]
36        \times (1-t^2)
37        \frac{J_1\left[qR\sin\theta \left(1-t^2\right)^{1/2}\right]}
38             {qR\sin\theta \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:** March 19, 2016
91"""
92from numpy import inf
93
94name = "capped_cylinder"
95title = "Right circular cylinder with spherical end caps and uniform SLD"
96description = """That is, a sphereocylinder
97    with end caps that have a radius larger than
98    that of the cylinder and the center of the
99    end cap radius lies within the cylinder.
100    Note: As the length of cylinder -->0,
101    it becomes a Convex Lens.
102    It must be that radius <(=) radius_cap.
103    [Parameters];
104    scale: volume fraction of spheres,
105    background:incoherent background,
106    radius: radius of the cylinder,
107    length: length of the cylinder,
108    radius_cap: radius of the semi-spherical cap,
109    sld: SLD of the capped cylinder,
110    sld_solvent: SLD of the solvent.
111"""
112category = "shape:cylinder"
113# pylint: disable=bad-whitespace, line-too-long
114#             ["name", "units", default, [lower, upper], "type", "description"],
115parameters = [["sld",         "1e-6/Ang^2", 4, [-inf, inf], "sld",    "Cylinder scattering length density"],
116              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",    "Solvent scattering length density"],
117              ["radius",      "Ang",       20, [0, inf],    "volume", "Cylinder radius"],
118
119              # TODO: use an expression for cap radius with fixed bounds.
120              # The current form requires cap radius R bigger than cylinder radius r.
121              # Could instead use R/r in [1,inf], r/R in [0,1], or the angle between
122              # cylinder and cap in [0,90].  The problem is similar for the barbell
123              # model.  Propose r/R in [0,1] in both cases, with the model specifying
124              # cylinder radius in the capped cylinder model and sphere radius in the
125              # barbell model.  This leads to the natural value of zero for no cap
126              # in the capped cylinder, and zero for no bar in the barbell model.  In
127              # both models, one would be a pill.
128              ["radius_cap", "Ang",     20, [0, inf],    "volume", "Cap radius"],
129              ["length",     "Ang",    400, [0, inf],    "volume", "Cylinder length"],
130              ["theta",      "degrees", 60, [-inf, inf], "orientation", "inclination angle"],
131              ["phi",        "degrees", 60, [-inf, inf], "orientation", "deflection angle"],
132             ]
133# pylint: enable=bad-whitespace, line-too-long
134
135source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "capped_cylinder.c"]
136
137demo = dict(scale=1, background=0,
138            sld=6, sld_solvent=1,
139            radius=260, radius_cap=290, length=290,
140            theta=30, phi=15,
141            radius_pd=.2, radius_pd_n=1,
142            radius_cap_pd=.2, radius_cap_pd_n=1,
143            length_pd=.2, length_pd_n=10,
144            theta_pd=15, theta_pd_n=45,
145            phi_pd=15, phi_pd_n=1)
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