source: sasmodels/sasmodels/models/barbell.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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1r"""
2Definition
3----------
4
5Calculates the scattering from a barbell-shaped cylinder.  Like
6:ref:`capped-cylinder`, this is a sphereocylinder with spherical end
7caps that have a radius larger than that of the cylinder, but with the center
8of the end cap radius lying outside of the cylinder. See the diagram for
9the details of the geometry and restrictions on parameter values.
10
11.. figure:: img/barbell_geometry.jpg
12
13    Barbell geometry, where $r$ is *radius*, $R$ is *radius_bell* and
14    $L$ is *length*. Since the end cap radius $R \geq r$ and by definition
15    for this geometry $h < 0$, $h$ is then defined by $r$ and $R$ as
16    $h = - \sqrt{R^2 - r^2}$
17
18The scattered intensity $I(q)$ is calculated as
19
20.. math::
21
22    I(q) = \frac{\Delta \rho^2}{V} \left<A^2(q,\alpha).sin(\alpha)\right>
23
24where the amplitude $A(q,\alpha)$ with the rod axis at angle $\alpha$ to $q$ is given as
25
26.. math::
27
28    A(q) =&\ \pi r^2L
29        \frac{\sin\left(\tfrac12 qL\cos\alpha\right)}
30             {\tfrac12 qL\cos\alpha}
31        \frac{2 J_1(qr\sin\alpha)}{qr\sin\alpha} \\
32        &\ + 4 \pi R^3 \int_{-h/R}^1 dt
33        \cos\left[ q\cos\alpha
34            \left(Rt + h + {\tfrac12} L\right)\right]
35        \times (1-t^2)
36        \frac{J_1\left[qR\sin\alpha \left(1-t^2\right)^{1/2}\right]}
37             {qR\sin\alpha \left(1-t^2\right)^{1/2}}
38
39The $\left<\ldots\right>$ brackets denote an average of the structure over
40all orientations. $\left<A^2(q,\alpha)\right>$ is then the form factor, $P(q)$.
41The scale factor is equivalent to the volume fraction of cylinders, each of
42volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length
43densities of the cylinder and the surrounding solvent.
44
45The volume of the barbell is
46
47.. math::
48
49    V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right)
50
51
52and its radius of gyration is
53
54.. math::
55
56    R_g^2 =&\ \left[ \tfrac{12}{5}R^5
57        + R^4\left(6h+\tfrac32 L\right)
58        + R^2\left(4h^2 + L^2 + 4Lh\right)
59        + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\
60        &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3
61        + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right]
62        \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1}
63
64.. note::
65    The requirement that $R \geq r$ is not enforced in the model! It is
66    up to you to restrict this during analysis.
67
68The 2D scattering intensity is calculated similar to the 2D cylinder model.
69
70.. figure:: img/cylinder_angle_definition.jpg
71
72    Definition of the angles for oriented 2D barbells.
73
74
75References
76----------
77
78.. [#] H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230
79.. [#] H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda
80   and errata)
81
82Authorship and Verification
83----------------------------
84
85* **Author:** NIST IGOR/DANSE **Date:** pre 2010
86* **Last Modified by:** Paul Butler **Date:** March 20, 2016
87* **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017
88"""
89from numpy import inf
90
91name = "barbell"
92title = "Cylinder with spherical end caps"
93description = """
94    Calculates the scattering from a barbell-shaped cylinder.
95    That is a sphereocylinder with spherical end caps that have a radius larger
96    than that of the cylinder and the center of the end cap radius lies outside
97    of the cylinder.
98    Note: As the length of cylinder(bar) -->0,it becomes a dumbbell. And when
99    rad_bar = rad_bell, it is a spherocylinder.
100    It must be that rad_bar <(=) rad_bell.
101"""
102category = "shape:cylinder"
103# pylint: disable=bad-whitespace, line-too-long
104#             ["name", "units", default, [lower, upper], "type","description"],
105parameters = [["sld",         "1e-6/Ang^2",   4, [-inf, inf], "sld",         "Barbell scattering length density"],
106              ["sld_solvent", "1e-6/Ang^2",   1, [-inf, inf], "sld",         "Solvent scattering length density"],
107              ["radius_bell", "Ang",         40, [0, inf],    "volume",      "Spherical bell radius"],
108              ["radius",      "Ang",         20, [0, inf],    "volume",      "Cylindrical bar radius"],
109              ["length",      "Ang",        400, [0, inf],    "volume",      "Cylinder bar length"],
110              ["theta",       "degrees",     60, [-inf, inf], "orientation", "In plane angle"],
111              ["phi",         "degrees",     60, [-inf, inf], "orientation", "Out of plane angle"],
112             ]
113# pylint: enable=bad-whitespace, line-too-long
114
115source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "barbell.c"]
116
117# parameters for demo
118demo = dict(scale=1, background=0,
119            sld=6, sld_solvent=1,
120            radius_bell=40, radius=20, length=400,
121            theta=60, phi=60,
122            radius_pd=.2, radius_pd_n=5,
123            length_pd=.2, length_pd_n=5,
124            theta_pd=15, theta_pd_n=0,
125            phi_pd=15, phi_pd_n=0,
126           )
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