source: sasmodels/sasmodels/models/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@…>, 11 months ago

provide method for testing Fq results. Refs #1202.

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1# cylinder model
2# Note: model title and parameter table are inserted automatically
3r"""
4
5For information about polarised and magnetic scattering, see
6the :ref:`magnetism` documentation.
7
8Definition
9----------
10
11The output of the 2D scattering intensity function for oriented cylinders is
12given by (Guinier, 1955)
13
14.. math::
15
16    P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha).sin(\alpha) + \text{background}
17
18where
19
20.. math::
21
22    F(q,\alpha) = 2 (\Delta \rho) V
23           \frac{\sin \left(\tfrac12 qL\cos\alpha \right)}
24                {\tfrac12 qL \cos \alpha}
25           \frac{J_1 \left(q R \sin \alpha\right)}{q R \sin \alpha}
26
27and $\alpha$ is the angle between the axis of the cylinder and $\vec q$, $V =\pi R^2L$
28is the volume of the cylinder, $L$ is the length of the cylinder, $R$ is the
29radius of the cylinder, and $\Delta\rho$ (contrast) is the scattering length
30density difference between the scatterer and the solvent. $J_1$ is the
31first order Bessel function.
32
33For randomly oriented particles:
34
35.. math::
36
37    F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}=\int_{0}^{1}{F^2(q,u)du}
38
39
40Numerical integration is simplified by a change of variable to $u = cos(\alpha)$ with
41$sin(\alpha)=\sqrt{1-u^2}$.
42
43The output of the 1D scattering intensity function for randomly oriented
44cylinders is thus given by
45
46.. math::
47
48    P(q) = \frac{\text{scale}}{V}
49        \int_0^{\pi/2} F^2(q,\alpha) \sin \alpha\ d\alpha + \text{background}
50
51
52NB: The 2nd virial coefficient of the cylinder is calculated based on the
53radius and length values, and used as the effective radius for $S(q)$
54when $P(q) \cdot S(q)$ is applied.
55
56For 2d scattering from oriented cylinders, we define the direction of the
57axis of the cylinder using two angles $\theta$ (note this is not the
58same as the scattering angle used in q) and $\phi$. Those angles
59are defined in :numref:`cylinder-angle-definition` , for further details see :ref:`orientation` .
60
61.. _cylinder-angle-definition:
62
63.. figure:: img/cylinder_angle_definition.png
64
65    Angles $\theta$ and $\phi$ orient the cylinder relative
66    to the beam line coordinates, where the beam is along the $z$ axis. Rotation $\theta$, initially
67    in the $xz$ plane, is carried out first, then rotation $\phi$ about the $z$ axis. Orientation distributions
68    are described as rotations about two perpendicular axes $\delta_1$ and $\delta_2$
69    in the frame of the cylinder itself, which when $\theta = \phi = 0$ are parallel to the $Y$ and $X$ axes.
70
71.. figure:: img/cylinder_angle_projection.png
72
73    Examples for oriented cylinders.
74
75The $\theta$ and $\phi$ parameters to orient the cylinder only appear in the model when fitting 2d data.
76
77Validation
78----------
79
80Validation of the code was done by comparing the output of the 1D model
81to the output of the software provided by the NIST (Kline, 2006).
82The implementation of the intensity for fully oriented cylinders was done
83by averaging over a uniform distribution of orientations using
84
85.. math::
86
87    P(q) = \int_0^{\pi/2} d\phi
88        \int_0^\pi p(\theta) P_0(q,\theta) \sin \theta\ d\theta
89
90
91where $p(\theta,\phi) = 1$ is the probability distribution for the orientation
92and $P_0(q,\theta)$ is the scattering intensity for the fully oriented
93system, and then comparing to the 1D result.
94
95References
96----------
97
98J. S. Pedersen, Adv. Colloid Interface Sci. 70, 171-210 (1997).
99G. Fournet, Bull. Soc. Fr. Mineral. Cristallogr. 74, 39-113 (1951).
100"""
101
102import numpy as np  # type: ignore
103from numpy import pi, inf  # type: ignore
104
105name = "cylinder"
106title = "Right circular cylinder with uniform scattering length density."
107description = """
108     f(q,alpha) = 2*(sld - sld_solvent)*V*sin(qLcos(alpha)/2))
109                /[qLcos(alpha)/2]*J1(qRsin(alpha))/[qRsin(alpha)]
110
111            P(q,alpha)= scale/V*f(q,alpha)^(2)+background
112            V: Volume of the cylinder
113            R: Radius of the cylinder
114            L: Length of the cylinder
115            J1: The bessel function
116            alpha: angle between the axis of the
117            cylinder and the q-vector for 1D
118            :the ouput is P(q)=scale/V*integral
119            from pi/2 to zero of...
120            f(q,alpha)^(2)*sin(alpha)*dalpha + background
121"""
122category = "shape:cylinder"
123
124#             [ "name", "units", default, [lower, upper], "type", "description"],
125parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld",
126               "Cylinder scattering length density"],
127              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
128               "Solvent scattering length density"],
129              ["radius", "Ang", 20, [0, inf], "volume",
130               "Cylinder radius"],
131              ["length", "Ang", 400, [0, inf], "volume",
132               "Cylinder length"],
133              ["theta", "degrees", 60, [-360, 360], "orientation",
134               "cylinder axis to beam angle"],
135              ["phi", "degrees", 60, [-360, 360], "orientation",
136               "rotation about beam"],
137             ]
138
139source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "cylinder.c"]
140have_Fq = True
141effective_radius_type = [
142    "equivalent sphere", "radius",
143    "half length", "half min dimension", "half max dimension", "half diagonal",
144    ]
145
146def random():
147    volume = 10**np.random.uniform(5, 12)
148    length = 10**np.random.uniform(-2, 2)*volume**0.333
149    radius = np.sqrt(volume/length/np.pi)
150    pars = dict(
151        #scale=1,
152        #background=0,
153        length=length,
154        radius=radius,
155    )
156    return pars
157
158
159# parameters for demo
160demo = dict(scale=1, background=0,
161            sld=6, sld_solvent=1,
162            radius=20, length=300,
163            theta=60, phi=60,
164            radius_pd=.2, radius_pd_n=9,
165            length_pd=.2, length_pd_n=10,
166            theta_pd=10, theta_pd_n=5,
167            phi_pd=10, phi_pd_n=5)
168
169# pylint: disable=bad-whitespace, line-too-long
170qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5)
171# After redefinition of angles, find new tests values.  Was 10 10 in old coords
172tests = [
173    [{}, 0.2, 0.042761386790780453],
174    [{}, [0.2], [0.042761386790780453]],
175    #  new coords
176    [{'theta':80.1534480601659, 'phi':10.1510817110481}, (qx, qy), 0.03514647218513852],
177    [{'theta':80.1534480601659, 'phi':10.1510817110481}, [(qx, qy)], [0.03514647218513852]],
178    # old coords
179    #[{'theta':10.0, 'phi':10.0}, (qx, qy), 0.03514647218513852],
180    #[{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.03514647218513852]],
181]
182del qx, qy  # not necessary to delete, but cleaner
183
184# Default radius and length
185radius, length = parameters[2][2], parameters[3][2]
186tests.extend([
187    ({'radius_effective_type': 0}, 0.1, None, None, 0., pi*radius**2*length, 1.0),
188    ({'radius_effective_type': 1}, 0.1, None, None, (0.75*radius**2*length)**(1./3.), None, None),
189    ({'radius_effective_type': 2}, 0.1, None, None, radius, None, None),
190    ({'radius_effective_type': 3}, 0.1, None, None, length/2., None, None),
191    ({'radius_effective_type': 4}, 0.1, None, None, min(radius, length/2.), None, None),
192    ({'radius_effective_type': 5}, 0.1, None, None, max(radius, length/2.), None, None),
193    ({'radius_effective_type': 6}, 0.1, None, None, np.sqrt(4*radius**2 + length**2)/2., None, None),
194])
195del radius, length
196# pylint: enable=bad-whitespace, line-too-long
197
198# ADDED by:  RKH  ON: 18Mar2016 renamed sld's etc
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