# source:sasmodels/sasmodels/models/cylinder.py@b1c40bee

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Last change on this file since b1c40bee was b1c40bee, checked in by wojciech, 6 years ago

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1# cylinder model
2# Note: model title and parameter table are inserted automatically
3r"""
4The form factor is normalized by the particle volume V = \piR^2L.
5For information about polarised and magnetic scattering, see
6the :doc:magnetic help <../sasgui/perspectives/fitting/mag_help> 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) + \text{background}
17
18where
19
20.. math::
21
22    F(q) = 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 $\vec q$, $V$
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
33To provide easy access to the orientation of the cylinder, we define the
34axis of the cylinder using two angles $\theta$ and $\phi$. Those angles
35are defined in :numref:cylinder-angle-definition.
36
37.. _cylinder-angle-definition:
38
39.. figure:: img/cylinder_angle_definition.jpg
40
41    Definition of the angles for oriented cylinders.
42
43.. figure:: img/cylinder_angle_projection.jpg
44
45    Examples of the angles for oriented cylinders against the detector plane.
46
47NB: The 2nd virial coefficient of the cylinder is calculated based on the
48radius and length values, and used as the effective radius for $S(q)$
49when $P(q) \cdot S(q)$ is applied.
50
51The output of the 1D scattering intensity function for randomly oriented
52cylinders is then given by
53
54.. math::
55
56    P(q) = \frac{\text{scale}}{V}
57        \int_0^{\pi/2} F^2(q,\alpha) \sin \alpha\ d\alpha + \text{background}
58
59The $\theta$ and $\phi$ parameters are not used for the 1D output.
60
61Validation
62----------
63
64Validation of the code was done by comparing the output of the 1D model
65to the output of the software provided by the NIST (Kline, 2006).
66The implementation of the intensity for fully oriented cylinders was done
67by averaging over a uniform distribution of orientations using
68
69.. math::
70
71    P(q) = \int_0^{\pi/2} d\phi
72        \int_0^\pi p(\theta, \phi) P_0(q,\alpha) \sin \theta\ d\theta
73
74
75where $p(\theta,\phi)$ is the probability distribution for the orientation
76and $P_0(q,\alpha)$ is the scattering intensity for the fully oriented
77system, and then comparing to the 1D result.
78
79References
80----------
81
82None
83
84"""
85
86import numpy as np  # type: ignore
87from numpy import pi, inf  # type: ignore
88
89name = "cylinder"
90title = "Right circular cylinder with uniform scattering length density."
91description = """
92     f(q,alpha) = 2*(sld - sld_solvent)*V*sin(qLcos(alpha)/2))
93                /[qLcos(alpha)/2]*J1(qRsin(alpha))/[qRsin(alpha)]
94
95            P(q,alpha)= scale/V*f(q,alpha)^(2)+background
96            V: Volume of the cylinder
97            R: Radius of the cylinder
98            L: Length of the cylinder
99            J1: The bessel function
100            alpha: angle between the axis of the
101            cylinder and the q-vector for 1D
102            :the ouput is P(q)=scale/V*integral
103            from pi/2 to zero of...
104            f(q,alpha)^(2)*sin(alpha)*dalpha + background
105"""
106category = "shape:cylinder"
107
108#             [ "name", "units", default, [lower, upper], "type", "description"],
109parameters = [["sld", "4e-6/Ang^2", 4, [-inf, inf], "sld",
110               "Cylinder scattering length density"],
111              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",
112               "Solvent scattering length density"],
113              ["radius", "Ang", 20, [0, inf], "volume",
115              ["length", "Ang", 400, [0, inf], "volume",
116               "Cylinder length"],
117              ["theta", "degrees", 60, [-inf, inf], "orientation",
118               "In plane angle"],
119              ["phi", "degrees", 60, [-inf, inf], "orientation",
120               "Out of plane angle"],
121             ]
122
123source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "cylinder.c"]
124
126    """
128    """
130    return 0.5 * (ddd) ** (1. / 3.)
131
132# parameters for demo
133demo = dict(scale=1, background=0,
134            sld=6, sld_solvent=1,
136            theta=60, phi=60,