source: sasmodels/sasmodels/models/cylinder.py @ b6422c7

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since b6422c7 was b297ba9, checked in by Paul Kienzle <pkienzle@…>, 5 years ago

lint

  • Property mode set to 100644
File size: 7.9 KB
Line 
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",
131               "Cylinder radius"],
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",
137               "rotation about beam"],
138             ]
139
140source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "cylinder.c"]
141have_Fq = True
142effective_radius_type = [
143    "excluded volume", "equivalent volume sphere", "radius",
144    "half length", "half min dimension", "half max dimension", "half diagonal",
145    ]
146
147def random():
148    """Return a random parameter set for the model."""
149    volume = 10**np.random.uniform(5, 12)
150    length = 10**np.random.uniform(-2, 2)*volume**0.333
151    radius = np.sqrt(volume/length/np.pi)
152    pars = dict(
153        #scale=1,
154        #background=0,
155        length=length,
156        radius=radius,
157    )
158    return pars
159
160
161# parameters for demo
162demo = dict(scale=1, background=0,
163            sld=6, sld_solvent=1,
164            radius=20, length=300,
165            theta=60, phi=60,
166            radius_pd=.2, radius_pd_n=9,
167            length_pd=.2, length_pd_n=10,
168            theta_pd=10, theta_pd_n=5,
169            phi_pd=10, phi_pd_n=5)
170
171# pylint: disable=bad-whitespace, line-too-long
172qx, qy = 0.2 * np.cos(2.5), 0.2 * np.sin(2.5)
173# After redefinition of angles, find new tests values.  Was 10 10 in old coords
174tests = [
175    [{}, 0.2, 0.042761386790780453],
176    [{}, [0.2], [0.042761386790780453]],
177    #  new coords
178    [{'theta':80.1534480601659, 'phi':10.1510817110481}, (qx, qy), 0.03514647218513852],
179    [{'theta':80.1534480601659, 'phi':10.1510817110481}, [(qx, qy)], [0.03514647218513852]],
180    # old coords
181    #[{'theta':10.0, 'phi':10.0}, (qx, qy), 0.03514647218513852],
182    #[{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.03514647218513852]],
183]
184del qx, qy  # not necessary to delete, but cleaner
185
186# Default radius and length
187def calc_volume(radius, length):
188    """Return form volume for cylinder."""
189    return pi*radius**2*length
190def calc_r_effs(radius, length):
191    """Return effective radii for modes 0-7 of cylinder."""
192    return [
193        0.,
194        0.5*(0.75*radius*(2.0*radius*length
195                          + (radius + length)*(pi*radius + length)))**(1./3.),
196        (0.75*radius**2*length)**(1./3.),
197        radius,
198        length/2.,
199        min(radius, length/2.),
200        max(radius, length/2.),
201        np.sqrt(4*radius**2 + length**2)/2.,
202    ]
203r_effs = calc_r_effs(parameters[2][2], parameters[3][2])
204cyl_vol = calc_volume(parameters[2][2], parameters[3][2])
205tests.extend([
206    ({'radius_effective_mode': 0}, 0.1, None, None, r_effs[0], cyl_vol, 1.0),
207    ({'radius_effective_mode': 1}, 0.1, None, None, r_effs[1], None, None),
208    ({'radius_effective_mode': 2}, 0.1, None, None, r_effs[2], None, None),
209    ({'radius_effective_mode': 3}, 0.1, None, None, r_effs[3], None, None),
210    ({'radius_effective_mode': 4}, 0.1, None, None, r_effs[4], None, None),
211    ({'radius_effective_mode': 5}, 0.1, None, None, r_effs[5], None, None),
212    ({'radius_effective_mode': 6}, 0.1, None, None, r_effs[6], None, None),
213    ({'radius_effective_mode': 7}, 0.1, None, None, r_effs[7], None, None),
214])
215del r_effs, cyl_vol
216# pylint: enable=bad-whitespace, line-too-long
217
218# ADDED by:  RKH  ON: 18Mar2016 renamed sld's etc
Note: See TracBrowser for help on using the repository browser.