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

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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.png
71
72    Definition of the angles for oriented 2D barbells.
73
74
75References
76----------
77
78.. [#] H Kaya, *J. Appl. Cryst.*, 37 (2004) 223-230
79.. [#] H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda
80   and errata)
81.. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659
82
83Source
84------
85
86barbell.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/barbell.py>_
87
88barbell.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/barbell.c>_
89
90Authorship and Verification
91----------------------------
92
93* **Author:** NIST IGOR/DANSE **Date:** pre 2010
95* **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017
96* **Source added by :** Steve King **Date:** March 25, 2019
97"""
98
99import numpy as np
100from numpy import inf, sin, cos, pi
101
102name = "barbell"
103title = "Cylinder with spherical end caps"
104description = """
105    Calculates the scattering from a barbell-shaped cylinder.
106    That is a sphereocylinder with spherical end caps that have a radius larger
107    than that of the cylinder and the center of the end cap radius lies outside
108    of the cylinder.
109    Note: As the length of cylinder(bar) -->0,it becomes a dumbbell. And when
112"""
113category = "shape:cylinder"
115#             ["name", "units", default, [lower, upper], "type","description"],
116parameters = [["sld",         "1e-6/Ang^2",   4, [-inf, inf], "sld",         "Barbell scattering length density"],
117              ["sld_solvent", "1e-6/Ang^2",   1, [-inf, inf], "sld",         "Solvent scattering length density"],
120              ["length",      "Ang",        400, [0, inf],    "volume",      "Cylinder bar length"],
121              ["theta",       "degrees",     60, [-360, 360], "orientation", "Barbell axis to beam angle"],
122              ["phi",         "degrees",     60, [-360, 360], "orientation", "Rotation about beam"],
123             ]
125
126source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "barbell.c"]
127have_Fq = True
129    "equivalent cylinder excluded volume", "equivalent volume sphere",
130    "radius", "half length", "half total length",
131    ]
132
133def random():
134    """Return a random parameter set for the model."""
135    # TODO: increase volume range once problem with bell radius is fixed
136    # The issue is that bell radii of more than about 200 fail at high q
137    volume = 10**np.random.uniform(7, 9)
138    bar_volume = 10**np.random.uniform(-4, -1)*volume
139    bell_volume = volume - bar_volume
140    bell_radius = (bell_volume/6)**0.3333  # approximate
142    bar_length = 10**np.random.uniform(0, 3)*min_bar
146    pars = dict(
147        #background=0,
150        length=bar_length,
151    )
152    return pars
153
154# parameters for demo
155demo = dict(scale=1, background=0,
156            sld=6, sld_solvent=1,
158            theta=60, phi=60,
160            length_pd=.2, length_pd_n=5,
161            theta_pd=15, theta_pd_n=0,
162            phi_pd=15, phi_pd_n=0,
163           )
164q = 0.1
165# april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct!
166qx = q*cos(pi/6.0)
167qy = q*sin(pi/6.0)
168tests = [
169    [{}, 0.075, 25.5691260532],
170    [{'theta':80., 'phi':10.}, (qx, qy), 3.04233067789],
171]
172del qx, qy  # not necessary to delete, but cleaner
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