source: sasmodels/sasmodels/models/barbell.py @ 99658f6

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 99658f6 was 99658f6, checked in by grethevj, 10 months ago

updated ER functions including cylinder excluded volume, to match 4.x

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Line 
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", "radius", "half length", "half total length",
122    ]
123
124def random():
125    # TODO: increase volume range once problem with bell radius is fixed
126    # The issue is that bell radii of more than about 200 fail at high q
127    volume = 10**np.random.uniform(7, 9)
128    bar_volume = 10**np.random.uniform(-4, -1)*volume
129    bell_volume = volume - bar_volume
130    bell_radius = (bell_volume/6)**0.3333  # approximate
131    min_bar = bar_volume/np.pi/bell_radius**2
132    bar_length = 10**np.random.uniform(0, 3)*min_bar
133    bar_radius = np.sqrt(bar_volume/bar_length/np.pi)
134    if bar_radius > bell_radius:
135        bell_radius, bar_radius = bar_radius, bell_radius
136    pars = dict(
137        #background=0,
138        radius_bell=bell_radius,
139        radius=bar_radius,
140        length=bar_length,
141    )
142    return pars
143
144# parameters for demo
145demo = dict(scale=1, background=0,
146            sld=6, sld_solvent=1,
147            radius_bell=40, radius=20, length=400,
148            theta=60, phi=60,
149            radius_pd=.2, radius_pd_n=5,
150            length_pd=.2, length_pd_n=5,
151            theta_pd=15, theta_pd_n=0,
152            phi_pd=15, phi_pd_n=0,
153           )
154q = 0.1
155# april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct!
156qx = q*cos(pi/6.0)
157qy = q*sin(pi/6.0)
158tests = [
159    [{}, 0.075, 25.5691260532],
160    [{'theta':80., 'phi':10.}, (qx, qy), 3.04233067789],
161]
162del qx, qy  # not necessary to delete, but cleaner
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