# source:sasmodels/sasmodels/models/hollow_cylinder.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, 11 months ago

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

• Property mode set to 100644
File size: 5.9 KB
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)
62L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949).
63
64Authorship and Verification
65----------------------------
66
67* **Author:** NIST IGOR/DANSE **Date:** pre 2010
68* **Last Modified by:** Paul Butler **Date:** September 06, 2018
69   (corrected VR calculation)
70* **Last Reviewed by:** Paul Butler **Date:** September 06, 2018
71"""
72from __future__ import division
73
74import numpy as np
75from numpy import pi, inf, sin, cos
76
77name = "hollow_cylinder"
78title = ""
79description = """
80P(q) = scale*<f*f>/Vol + background, where f is the scattering amplitude.
82thickness = the thickness of shell
83length = the total length of the cylinder
84sld = SLD of the shell
85sld_solvent = SLD of the solvent
86background = incoherent background
87"""
88category = "shape:cylinder"
89# pylint: disable=bad-whitespace, line-too-long
90#   ["name", "units", default, [lower, upper], "type","description"],
91parameters = [
92    ["radius",      "Ang",     20.0, [0, inf],    "volume",      "Cylinder core radius"],
93    ["thickness",   "Ang",     10.0, [0, inf],    "volume",      "Cylinder wall thickness"],
94    ["length",      "Ang",    400.0, [0, inf],    "volume",      "Cylinder total length"],
95    ["sld",         "1e-6/Ang^2",  6.3, [-inf, inf], "sld",         "Cylinder sld"],
96    ["sld_solvent", "1e-6/Ang^2",  1,   [-inf, inf], "sld",         "Solvent sld"],
97    ["theta",       "degrees", 90,   [-360, 360], "orientation", "Cylinder axis to beam angle"],
98    ["phi",         "degrees",  0,   [-360, 360], "orientation", "Rotation about beam"],
99    ]
100# pylint: enable=bad-whitespace, line-too-long
101
102source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "hollow_cylinder.c"]
103have_Fq = True
105    "excluded volume", "equivalent outer volume sphere", "outer radius", "half length",
106    "half outer min dimension", "half outer max dimension",
107    "half outer diagonal",
108    ]
109
110def random():
111    length = 10**np.random.uniform(1, 4.7)
112    outer_radius = 10**np.random.uniform(1, 4.7)
113    # Use a distribution with a preference for thin shell or thin core
114    # Avoid core,shell radii < 1
115    thickness = np.random.beta(0.5, 0.5)*(outer_radius-2) + 1
117    pars = dict(
118        length=length,
120        thickness=thickness,
121    )
122    return pars
123
124# parameters for demo
125demo = dict(scale=1.0, background=0.0, length=400.0, radius=20.0,
126            thickness=10, sld=6.3, sld_solvent=1, theta=90, phi=0,
127            thickness_pd=0.2, thickness_pd_n=9,
128            length_pd=.2, length_pd_n=10,
130            theta_pd=10, theta_pd_n=5,
131           )
132q = 0.1
133# april 6 2017, rkh added a 2d unit test, assume correct!
134qx = q*cos(pi/6.0)
135qy = q*sin(pi/6.0)
137thickness = parameters
138length = parameters
139# Parameters for unit tests
140tests = [
141    [{}, 0.00005, 1764.926],
142    [{}, 0.1, None, None,