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

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since b297ba9 was b297ba9, checked in by Paul Kienzle <pkienzle@…>, 7 months ago

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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) 37 223-230
79.. [#] H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda
80   and errata)
81L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949).
82
83Authorship and Verification
84----------------------------
85
86* **Author:** NIST IGOR/DANSE **Date:** pre 2010
87* **Last Modified by:** Paul Butler **Date:** March 20, 2016
88* **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017
89"""
90
91import numpy as np
92from numpy import inf, sin, cos, pi
93
94name = "barbell"
95title = "Cylinder with spherical end caps"
96description = """
97    Calculates the scattering from a barbell-shaped cylinder.
98    That is a sphereocylinder with spherical end caps that have a radius larger
99    than that of the cylinder and the center of the end cap radius lies outside
100    of the cylinder.
101    Note: As the length of cylinder(bar) -->0,it becomes a dumbbell. And when
102    rad_bar = rad_bell, it is a spherocylinder.
103    It must be that rad_bar <(=) rad_bell.
104"""
105category = "shape:cylinder"
106# pylint: disable=bad-whitespace, line-too-long
107#             ["name", "units", default, [lower, upper], "type","description"],
108parameters = [["sld",         "1e-6/Ang^2",   4, [-inf, inf], "sld",         "Barbell scattering length density"],
109              ["sld_solvent", "1e-6/Ang^2",   1, [-inf, inf], "sld",         "Solvent scattering length density"],
110              ["radius_bell", "Ang",         40, [0, inf],    "volume",      "Spherical bell radius"],
111              ["radius",      "Ang",         20, [0, inf],    "volume",      "Cylindrical bar radius"],
112              ["length",      "Ang",        400, [0, inf],    "volume",      "Cylinder bar length"],
113              ["theta",       "degrees",     60, [-360, 360], "orientation", "Barbell axis to beam angle"],
114              ["phi",         "degrees",     60, [-360, 360], "orientation", "Rotation about beam"],
115             ]
116# pylint: enable=bad-whitespace, line-too-long
117
118source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "barbell.c"]
119have_Fq = True
120effective_radius_type = [
121    "equivalent cylinder excluded volume", "equivalent volume sphere",
122    "radius", "half length", "half total length",
123    ]
124
125def random():
126    """Return a random parameter set for the model."""
127    # TODO: increase volume range once problem with bell radius is fixed
128    # The issue is that bell radii of more than about 200 fail at high q
129    volume = 10**np.random.uniform(7, 9)
130    bar_volume = 10**np.random.uniform(-4, -1)*volume
131    bell_volume = volume - bar_volume
132    bell_radius = (bell_volume/6)**0.3333  # approximate
133    min_bar = bar_volume/np.pi/bell_radius**2
134    bar_length = 10**np.random.uniform(0, 3)*min_bar
135    bar_radius = np.sqrt(bar_volume/bar_length/np.pi)
136    if bar_radius > bell_radius:
137        bell_radius, bar_radius = bar_radius, bell_radius
138    pars = dict(
139        #background=0,
140        radius_bell=bell_radius,
141        radius=bar_radius,
142        length=bar_length,
143    )
144    return pars
145
146# parameters for demo
147demo = dict(scale=1, background=0,
148            sld=6, sld_solvent=1,
149            radius_bell=40, radius=20, length=400,
150            theta=60, phi=60,
151            radius_pd=.2, radius_pd_n=5,
152            length_pd=.2, length_pd_n=5,
153            theta_pd=15, theta_pd_n=0,
154            phi_pd=15, phi_pd_n=0,
155           )
156q = 0.1
157# april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct!
158qx = q*cos(pi/6.0)
159qy = q*sin(pi/6.0)
160tests = [
161    [{}, 0.075, 25.5691260532],
162    [{'theta':80., 'phi':10.}, (qx, qy), 3.04233067789],
163]
164del qx, qy  # not necessary to delete, but cleaner
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