source: sasmodels/sasmodels/models/cylinder.py @ c1e44e5

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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
98.. [#] J. Pedersen, *Adv. Colloid Interface Sci.*, 70 (1997) 171-210
99.. [#] G. Fournet, *Bull. Soc. Fr. Mineral. Cristallogr.*, 74 (1951) 39-113
100.. [#] L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659
101
102Authorship and Verification
103----------------------------
104
105* **Author:**
106* **Last Modified by:**
107* **Last Reviewed by:**
108"""
109
110import numpy as np  # type: ignore
111from numpy import pi, inf  # type: ignore
112
113name = "cylinder"
114title = "Right circular cylinder with uniform scattering length density."
115description = """
116     f(q,alpha) = 2*(sld - sld_solvent)*V*sin(qLcos(alpha)/2))
117                /[qLcos(alpha)/2]*J1(qRsin(alpha))/[qRsin(alpha)]
118
119            P(q,alpha)= scale/V*f(q,alpha)^(2)+background
120            V: Volume of the cylinder
121            R: Radius of the cylinder
122            L: Length of the cylinder
123            J1: The bessel function
124            alpha: angle between the axis of the
125            cylinder and the q-vector for 1D
126            :the ouput is P(q)=scale/V*integral
127            from pi/2 to zero of...
128            f(q,alpha)^(2)*sin(alpha)*dalpha + background
129"""
130category = "shape:cylinder"
131
132#             [ "name", "units", default, [lower, upper], "type", "description"],
133parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld",
134               "Cylinder scattering length density"],
135              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
136               "Solvent scattering length density"],
137              ["radius", "Ang", 20, [0, inf], "volume",
138               "Cylinder radius"],
139              ["length", "Ang", 400, [0, inf], "volume",
140               "Cylinder length"],
141              ["theta", "degrees", 60, [-360, 360], "orientation",
142               "cylinder axis to beam angle"],
143              ["phi", "degrees", 60, [-360, 360], "orientation",
144               "rotation about beam"],
145             ]
146
147source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "cylinder.c"]
148have_Fq = True
149effective_radius_type = [
150    "excluded volume", "equivalent volume sphere", "radius",
151    "half length", "half min dimension", "half max dimension", "half diagonal",
152    ]
153
154def random():
155    """Return a random parameter set for the model."""
156    volume = 10**np.random.uniform(5, 12)
157    length = 10**np.random.uniform(-2, 2)*volume**0.333
158    radius = np.sqrt(volume/length/np.pi)
159    pars = dict(
160        #scale=1,
161        #background=0,
162        length=length,
163        radius=radius,
164    )
165    return pars
166
167
168# parameters for demo
169demo = dict(scale=1, background=0,
170            sld=6, sld_solvent=1,
171            radius=20, length=300,
172            theta=60, phi=60,
173            radius_pd=.2, radius_pd_n=9,
174            length_pd=.2, length_pd_n=10,
175            theta_pd=10, theta_pd_n=5,
176            phi_pd=10, phi_pd_n=5)
177
178# pylint: disable=bad-whitespace, line-too-long
179qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5)
180# After redefinition of angles, find new tests values.  Was 10 10 in old coords
181tests = [
182    [{}, 0.2, 0.042761386790780453],
183    [{}, [0.2], [0.042761386790780453]],
184    #  new coords
185    [{'theta':80.1534480601659, 'phi':10.1510817110481}, (qx, qy), 0.03514647218513852],
186    [{'theta':80.1534480601659, 'phi':10.1510817110481}, [(qx, qy)], [0.03514647218513852]],
187    # old coords
188    #[{'theta':10.0, 'phi':10.0}, (qx, qy), 0.03514647218513852],
189    #[{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.03514647218513852]],
190]
191del qx, qy  # not necessary to delete, but cleaner
192
193# Default radius and length
194def calc_volume(radius, length):
195    """Return form volume for cylinder."""
196    return pi*radius**2*length
197def calc_r_effs(radius, length):
198    """Return effective radii for modes 0-7 of cylinder."""
199    return [
200        0.,
201        0.5*(0.75*radius*(2.0*radius*length
202                          + (radius + length)*(pi*radius + length)))**(1./3.),
203        (0.75*radius**2*length)**(1./3.),
204        radius,
205        length/2.,
206        min(radius, length/2.),
207        max(radius, length/2.),
208        np.sqrt(4*radius**2 + length**2)/2.,
209    ]
210r_effs = calc_r_effs(parameters[2][2], parameters[3][2])
211cyl_vol = calc_volume(parameters[2][2], parameters[3][2])
212tests.extend([
213    ({'radius_effective_mode': 0}, 0.1, None, None, r_effs[0], cyl_vol, 1.0),
214    ({'radius_effective_mode': 1}, 0.1, None, None, r_effs[1], None, None),
215    ({'radius_effective_mode': 2}, 0.1, None, None, r_effs[2], None, None),
216    ({'radius_effective_mode': 3}, 0.1, None, None, r_effs[3], None, None),
217    ({'radius_effective_mode': 4}, 0.1, None, None, r_effs[4], None, None),
218    ({'radius_effective_mode': 5}, 0.1, None, None, r_effs[5], None, None),
219    ({'radius_effective_mode': 6}, 0.1, None, None, r_effs[6], None, None),
220    ({'radius_effective_mode': 7}, 0.1, None, None, r_effs[7], None, None),
221])
222del r_effs, cyl_vol
223# pylint: enable=bad-whitespace, line-too-long
224
225# ADDED by:  RKH  ON: 18Mar2016 renamed sld's etc
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