source: sasmodels/sasmodels/models/barbell.py @ 5024a56

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 5024a56 was ee60aa7, checked in by Paul Kienzle <pkienzle@…>, 6 years ago

clean up effective radius functions; improve mono_gauss_coil accuracy; start moving VR into C

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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 = [
120    "equivalent sphere", "radius", "half length", "half total length",
121    ]
122
123def random():
124    # TODO: increase volume range once problem with bell radius is fixed
125    # The issue is that bell radii of more than about 200 fail at high q
126    volume = 10**np.random.uniform(7, 9)
127    bar_volume = 10**np.random.uniform(-4, -1)*volume
128    bell_volume = volume - bar_volume
129    bell_radius = (bell_volume/6)**0.3333  # approximate
130    min_bar = bar_volume/np.pi/bell_radius**2
131    bar_length = 10**np.random.uniform(0, 3)*min_bar
132    bar_radius = np.sqrt(bar_volume/bar_length/np.pi)
133    if bar_radius > bell_radius:
134        bell_radius, bar_radius = bar_radius, bell_radius
135    pars = dict(
136        #background=0,
137        radius_bell=bell_radius,
138        radius=bar_radius,
139        length=bar_length,
140    )
141    return pars
142
143# parameters for demo
144demo = dict(scale=1, background=0,
145            sld=6, sld_solvent=1,
146            radius_bell=40, radius=20, length=400,
147            theta=60, phi=60,
148            radius_pd=.2, radius_pd_n=5,
149            length_pd=.2, length_pd_n=5,
150            theta_pd=15, theta_pd_n=0,
151            phi_pd=15, phi_pd_n=0,
152           )
153q = 0.1
154# april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct!
155qx = q*cos(pi/6.0)
156qy = q*sin(pi/6.0)
157tests = [
158    [{}, 0.075, 25.5691260532],
159    [{'theta':80., 'phi':10.}, (qx, qy), 3.04233067789],
160]
161del qx, qy  # not necessary to delete, but cleaner
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