source: sasmodels/sasmodels/models/capped_cylinder.py @ b297ba9

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
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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.png
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)
84L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949).
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
94import numpy as np
95from numpy import inf, sin, cos, pi
96
97name = "capped_cylinder"
98title = "Right circular cylinder with spherical end caps and uniform SLD"
99description = """That is, a sphereocylinder
100    with end caps that have a radius larger than
101    that of the cylinder and the center of the
102    end cap radius lies within the cylinder.
103    Note: As the length of cylinder -->0,
104    it becomes a Convex Lens.
105    It must be that radius <(=) radius_cap.
106    [Parameters];
107    scale: volume fraction of spheres,
108    background:incoherent background,
109    radius: radius of the cylinder,
110    length: length of the cylinder,
111    radius_cap: radius of the semi-spherical cap,
112    sld: SLD of the capped cylinder,
113    sld_solvent: SLD of the solvent.
114"""
115category = "shape:cylinder"
116# pylint: disable=bad-whitespace, line-too-long
117#             ["name", "units", default, [lower, upper], "type", "description"],
118parameters = [["sld",         "1e-6/Ang^2", 4, [-inf, inf], "sld",    "Cylinder scattering length density"],
119              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",    "Solvent scattering length density"],
120              ["radius",      "Ang",       20, [0, inf],    "volume", "Cylinder radius"],
121
122              # TODO: use an expression for cap radius with fixed bounds.
123              # The current form requires cap radius R bigger than cylinder radius r.
124              # Could instead use R/r in [1,inf], r/R in [0,1], or the angle between
125              # cylinder and cap in [0,90].  The problem is similar for the barbell
126              # model.  Propose r/R in [0,1] in both cases, with the model specifying
127              # cylinder radius in the capped cylinder model and sphere radius in the
128              # barbell model.  This leads to the natural value of zero for no cap
129              # in the capped cylinder, and zero for no bar in the barbell model.  In
130              # both models, one would be a pill.
131              ["radius_cap", "Ang",     20, [0, inf],    "volume", "Cap radius"],
132              ["length",     "Ang",    400, [0, inf],    "volume", "Cylinder length"],
133              ["theta",      "degrees", 60, [-360, 360], "orientation", "cylinder axis to beam angle"],
134              ["phi",        "degrees", 60, [-360, 360], "orientation", "rotation about beam"],
135             ]
136# pylint: enable=bad-whitespace, line-too-long
137
138source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "capped_cylinder.c"]
139have_Fq = True
140effective_radius_type = [
141    "equivalent cylinder excluded volume", "equivalent volume sphere",
142    "radius", "half length", "half total length",
143    ]
144
145def random():
146    """Return a random parameter set for the model."""
147    # TODO: increase volume range once problem with bell radius is fixed
148    # The issue is that bell radii of more than about 200 fail at high q
149    volume = 10**np.random.uniform(7, 9)
150    bar_volume = 10**np.random.uniform(-4, -1)*volume
151    bell_volume = volume - bar_volume
152    bell_radius = (bell_volume/6)**0.3333  # approximate
153    min_bar = bar_volume/np.pi/bell_radius**2
154    bar_length = 10**np.random.uniform(0, 3)*min_bar
155    bar_radius = np.sqrt(bar_volume/bar_length/np.pi)
156    if bar_radius > bell_radius:
157        bell_radius, bar_radius = bar_radius, bell_radius
158    pars = dict(
159        #background=0,
160        radius_cap=bell_radius,
161        radius=bar_radius,
162        length=bar_length,
163    )
164    return pars
165
166
167demo = dict(scale=1, background=0,
168            sld=6, sld_solvent=1,
169            radius=260, radius_cap=290, length=290,
170            theta=30, phi=15,
171            radius_pd=.2, radius_pd_n=1,
172            radius_cap_pd=.2, radius_cap_pd_n=1,
173            length_pd=.2, length_pd_n=10,
174            theta_pd=15, theta_pd_n=45,
175            phi_pd=15, phi_pd_n=1)
176q = 0.1
177# april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct!
178qx = q*cos(pi/6.0)
179qy = q*sin(pi/6.0)
180tests = [
181    [{}, 0.075, 26.0698570695],
182    [{'theta':80., 'phi':10.}, (qx, qy), 0.561811990502],
183]
184del qx, qy  # not necessary to delete, but cleaner
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