source: sasmodels/sasmodels/models/barbell.py @ d277229

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since d277229 was d277229, checked in by grethevj, 6 years ago

Models updated to include choices for effective interaction radii

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