source: sasmodels/sasmodels/models/hollow_cylinder.py @ 304c775

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 304c775 was 304c775, checked in by Paul Kienzle <pkienzle@…>, 8 months ago

provide method for testing Fq results. Refs #1202.

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Line 
1r"""
2Definition
3----------
4
5This model provides the form factor, $P(q)$, for a monodisperse hollow right
6angle circular cylinder (rigid tube) where the The inside and outside of the
7hollow cylinder are assumed to have the same SLD and the form factor is thus
8normalized by the volume of the tube (i.e. not by the total cylinder volume).
9
10.. math::
11
12    P(q) = \text{scale} \left<F^2\right>/V_\text{shell} + \text{background}
13
14where the averaging $\left<\ldots\right>$ is applied only for the 1D
15calculation. If Intensity is given on an absolute scale, the scale factor here
16is the volume fraction of the shell.  This differs from
17the :ref:`core-shell-cylinder` in that, in that case, scale is the volume
18fraction of the entire cylinder (core+shell). The application might be for a
19bilayer which wraps into a hollow tube and the volume fraction of material is
20all in the shell, whereas the :ref:`core-shell-cylinder` model might be used for
21a cylindrical micelle where the tails in the core have a different SLD than the
22headgroups (in the shell) and the volume fraction of material comes fromm the
23whole cyclinder.  NOTE: the hollow_cylinder represents a tube whereas the
24core_shell_cylinder includes a shell layer covering the ends (end caps) as well.
25
26
27The 1D scattering intensity is calculated in the following way (Guinier, 1955)
28
29.. math::
30
31    P(q)           &= (\text{scale})V_\text{shell}\Delta\rho^2
32            \int_0^{1}\Psi^2
33            \left[q_z, R_\text{outer}(1-x^2)^{1/2},
34                       R_\text{core}(1-x^2)^{1/2}\right]
35            \left[\frac{\sin(qHx)}{qHx}\right]^2 dx \\
36    \Psi[q,y,z]    &= \frac{1}{1-\gamma^2}
37            \left[ \Lambda(qy) - \gamma^2\Lambda(qz) \right] \\
38    \Lambda(a)     &= 2 J_1(a) / a \\
39    \gamma         &= R_\text{core} / R_\text{outer} \\
40    V_\text{shell} &= \pi \left(R_\text{outer}^2 - R_\text{core}^2 \right)L \\
41    J_1(x)         &= (\sin(x)-x\cdot \cos(x)) / x^2
42
43where *scale* is a scale factor, $H = L/2$ and $J_1$ is the 1st order
44Bessel function.
45
46**NB**: The 2nd virial coefficient of the cylinder is calculated
47based on the outer radius and full length, which give an the effective radius
48for structure factor $S(q)$ when $P(q) \cdot S(q)$ is applied.
49
50In the parameters,the *radius* is $R_\text{core}$ while *thickness*
51is $R_\text{outer} - R_\text{core}$.
52
53To provide easy access to the orientation of the core-shell cylinder, we define
54the axis of the cylinder using two angles $\theta$ and $\phi$
55(see :ref:`cylinder model <cylinder-angle-definition>`).
56
57References
58----------
59
60.. [#] L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and
61   Neutron Scattering*, Plenum Press, New York, (1987)
62
63Authorship and Verification
64----------------------------
65
66* **Author:** NIST IGOR/DANSE **Date:** pre 2010
67* **Last Modified by:** Paul Butler **Date:** September 06, 2018
68   (corrected VR calculation)
69* **Last Reviewed by:** Paul Butler **Date:** September 06, 2018
70"""
71from __future__ import division
72
73import numpy as np
74from numpy import pi, inf, sin, cos
75
76name = "hollow_cylinder"
77title = ""
78description = """
79P(q) = scale*<f*f>/Vol + background, where f is the scattering amplitude.
80radius = the radius of core
81thickness = the thickness of shell
82length = the total length of the cylinder
83sld = SLD of the shell
84sld_solvent = SLD of the solvent
85background = incoherent background
86"""
87category = "shape:cylinder"
88# pylint: disable=bad-whitespace, line-too-long
89#   ["name", "units", default, [lower, upper], "type","description"],
90parameters = [
91    ["radius",      "Ang",     20.0, [0, inf],    "volume",      "Cylinder core radius"],
92    ["thickness",   "Ang",     10.0, [0, inf],    "volume",      "Cylinder wall thickness"],
93    ["length",      "Ang",    400.0, [0, inf],    "volume",      "Cylinder total length"],
94    ["sld",         "1e-6/Ang^2",  6.3, [-inf, inf], "sld",         "Cylinder sld"],
95    ["sld_solvent", "1e-6/Ang^2",  1,   [-inf, inf], "sld",         "Solvent sld"],
96    ["theta",       "degrees", 90,   [-360, 360], "orientation", "Cylinder axis to beam angle"],
97    ["phi",         "degrees",  0,   [-360, 360], "orientation", "Rotation about beam"],
98    ]
99# pylint: enable=bad-whitespace, line-too-long
100
101source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "hollow_cylinder.c"]
102have_Fq = True
103effective_radius_type = [
104    "equivalent sphere", "outer radius", "half length",
105    "half outer min dimension", "half outer max dimension",
106    "half outer diagonal",
107    ]
108
109def random():
110    length = 10**np.random.uniform(1, 4.7)
111    outer_radius = 10**np.random.uniform(1, 4.7)
112    # Use a distribution with a preference for thin shell or thin core
113    # Avoid core,shell radii < 1
114    thickness = np.random.beta(0.5, 0.5)*(outer_radius-2) + 1
115    radius = outer_radius - thickness
116    pars = dict(
117        length=length,
118        radius=radius,
119        thickness=thickness,
120    )
121    return pars
122
123# parameters for demo
124demo = dict(scale=1.0, background=0.0, length=400.0, radius=20.0,
125            thickness=10, sld=6.3, sld_solvent=1, theta=90, phi=0,
126            thickness_pd=0.2, thickness_pd_n=9,
127            length_pd=.2, length_pd_n=10,
128            radius_pd=.2, radius_pd_n=9,
129            theta_pd=10, theta_pd_n=5,
130           )
131q = 0.1
132# april 6 2017, rkh added a 2d unit test, assume correct!
133qx = q*cos(pi/6.0)
134qy = q*sin(pi/6.0)
135radius = parameters[0][2]
136thickness = parameters[1][2]
137length = parameters[2][2]
138# Parameters for unit tests
139tests = [
140    [{}, 0.00005, 1764.926],
141    [{}, 0.1, None, None,
142     (3./4*(radius+thickness)**2*length)**(1./3),  # R_eff from volume
143     pi*((radius+thickness)**2-radius**2)*length,  # shell volume
144     (radius+thickness)**2/((radius+thickness)**2 - radius**2), # form:shell ratio
145    ],
146    [{}, 0.001, 1756.76],
147    [{}, (qx, qy), 2.36885476192],
148]
149del qx, qy  # not necessary to delete, but cleaner
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