# source:sasmodels/sasmodels/models/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, 13 months ago

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

• Property mode set to 100644
File size: 7.4 KB
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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).
100L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949).
101"""
102
103import numpy as np  # type: ignore
104from numpy import pi, inf  # type: ignore
105
106name = "cylinder"
107title = "Right circular cylinder with uniform scattering length density."
108description = """
109     f(q,alpha) = 2*(sld - sld_solvent)*V*sin(qLcos(alpha)/2))
110                /[qLcos(alpha)/2]*J1(qRsin(alpha))/[qRsin(alpha)]
111
112            P(q,alpha)= scale/V*f(q,alpha)^(2)+background
113            V: Volume of the cylinder
114            R: Radius of the cylinder
115            L: Length of the cylinder
116            J1: The bessel function
117            alpha: angle between the axis of the
118            cylinder and the q-vector for 1D
119            :the ouput is P(q)=scale/V*integral
120            from pi/2 to zero of...
121            f(q,alpha)^(2)*sin(alpha)*dalpha + background
122"""
123category = "shape:cylinder"
124
125#             [ "name", "units", default, [lower, upper], "type", "description"],
126parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld",
127               "Cylinder scattering length density"],
128              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
129               "Solvent scattering length density"],
130              ["radius", "Ang", 20, [0, inf], "volume",
132              ["length", "Ang", 400, [0, inf], "volume",
133               "Cylinder length"],
134              ["theta", "degrees", 60, [-360, 360], "orientation",
135               "cylinder axis to beam angle"],
136              ["phi", "degrees", 60, [-360, 360], "orientation",
138             ]
139
140source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "cylinder.c"]
141have_Fq = True
143    "excluded volume", "equivalent volume sphere", "radius",
144    "half length", "half min dimension", "half max dimension", "half diagonal",
145    ]
146
147def random():
148    volume = 10**np.random.uniform(5, 12)
149    length = 10**np.random.uniform(-2, 2)*volume**0.333
151    pars = dict(
152        #scale=1,
153        #background=0,
154        length=length,
156    )
157    return pars
158
159
160# parameters for demo
161demo = dict(scale=1, background=0,
162            sld=6, sld_solvent=1,
164            theta=60, phi=60,
166            length_pd=.2, length_pd_n=10,
167            theta_pd=10, theta_pd_n=5,
168            phi_pd=10, phi_pd_n=5)
169
171qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5)
172# After redefinition of angles, find new tests values.  Was 10 10 in old coords
173tests = [
174    [{}, 0.2, 0.042761386790780453],
175    [{}, [0.2], [0.042761386790780453]],
176    #  new coords
177    [{'theta':80.1534480601659, 'phi':10.1510817110481}, (qx, qy), 0.03514647218513852],
178    [{'theta':80.1534480601659, 'phi':10.1510817110481}, [(qx, qy)], [0.03514647218513852]],
179    # old coords
180    #[{'theta':10.0, 'phi':10.0}, (qx, qy), 0.03514647218513852],
181    #[{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.03514647218513852]],
182]
183del qx, qy  # not necessary to delete, but cleaner
184
187tests.extend([