1 | # cylinder model |
---|
2 | # Note: model title and parameter table are inserted automatically |
---|
3 | r""" |
---|
4 | |
---|
5 | For information about polarised and magnetic scattering, see |
---|
6 | the :ref:`magnetism` documentation. |
---|
7 | |
---|
8 | Definition |
---|
9 | ---------- |
---|
10 | |
---|
11 | The output of the 2D scattering intensity function for oriented cylinders is |
---|
12 | given by (Guinier, 1955) |
---|
13 | |
---|
14 | .. math:: |
---|
15 | |
---|
16 | P(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha).sin(\alpha) + \text{background} |
---|
17 | |
---|
18 | where |
---|
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 | |
---|
27 | and $\alpha$ is the angle between the axis of the cylinder and $\vec q$, $V =\pi R^2L$ |
---|
28 | is the volume of the cylinder, $L$ is the length of the cylinder, $R$ is the |
---|
29 | radius of the cylinder, and $\Delta\rho$ (contrast) is the scattering length |
---|
30 | density difference between the scatterer and the solvent. $J_1$ is the |
---|
31 | first order Bessel function. |
---|
32 | |
---|
33 | For 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 | |
---|
40 | Numerical integration is simplified by a change of variable to $u = cos(\alpha)$ with |
---|
41 | $sin(\alpha)=\sqrt{1-u^2}$. |
---|
42 | |
---|
43 | The output of the 1D scattering intensity function for randomly oriented |
---|
44 | cylinders 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 | |
---|
52 | NB: The 2nd virial coefficient of the cylinder is calculated based on the |
---|
53 | radius and length values, and used as the effective radius for $S(q)$ |
---|
54 | when $P(q) \cdot S(q)$ is applied. |
---|
55 | |
---|
56 | For 2d scattering from oriented cylinders, we define the direction of the |
---|
57 | axis of the cylinder using two angles $\theta$ (note this is not the |
---|
58 | same as the scattering angle used in q) and $\phi$. Those angles |
---|
59 | are 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 | |
---|
75 | The $\theta$ and $\phi$ parameters to orient the cylinder only appear in the model when fitting 2d data. |
---|
76 | |
---|
77 | Validation |
---|
78 | ---------- |
---|
79 | |
---|
80 | Validation of the code was done by comparing the output of the 1D model |
---|
81 | to the output of the software provided by the NIST (Kline, 2006). |
---|
82 | The implementation of the intensity for fully oriented cylinders was done |
---|
83 | by 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 | |
---|
91 | where $p(\theta,\phi) = 1$ is the probability distribution for the orientation |
---|
92 | and $P_0(q,\theta)$ is the scattering intensity for the fully oriented |
---|
93 | system, and then comparing to the 1D result. |
---|
94 | |
---|
95 | References |
---|
96 | ---------- |
---|
97 | |
---|
98 | J. S. Pedersen, Adv. Colloid Interface Sci. 70, 171-210 (1997). |
---|
99 | G. Fournet, Bull. Soc. Fr. Mineral. Cristallogr. 74, 39-113 (1951). |
---|
100 | """ |
---|
101 | |
---|
102 | import numpy as np # type: ignore |
---|
103 | from numpy import pi, inf # type: ignore |
---|
104 | |
---|
105 | name = "cylinder" |
---|
106 | title = "Right circular cylinder with uniform scattering length density." |
---|
107 | description = """ |
---|
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 | """ |
---|
122 | category = "shape:cylinder" |
---|
123 | |
---|
124 | # [ "name", "units", default, [lower, upper], "type", "description"], |
---|
125 | parameters = [["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 | |
---|
139 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "cylinder.c"] |
---|
140 | |
---|
141 | def ER(radius, length): |
---|
142 | """ |
---|
143 | Return equivalent radius (ER) |
---|
144 | """ |
---|
145 | ddd = 0.75 * radius * (2 * radius * length + (length + radius) * (length + pi * radius)) |
---|
146 | return 0.5 * (ddd) ** (1. / 3.) |
---|
147 | |
---|
148 | def random(): |
---|
149 | import numpy as np |
---|
150 | V = 10**np.random.uniform(5, 12) |
---|
151 | length = 10**np.random.uniform(-2, 2)*V**0.333 |
---|
152 | radius = np.sqrt(V/length/np.pi) |
---|
153 | pars = dict( |
---|
154 | #scale=1, |
---|
155 | #background=0, |
---|
156 | length=length, |
---|
157 | radius=radius, |
---|
158 | ) |
---|
159 | return pars |
---|
160 | |
---|
161 | |
---|
162 | # parameters for demo |
---|
163 | demo = dict(scale=1, background=0, |
---|
164 | sld=6, sld_solvent=1, |
---|
165 | radius=20, length=300, |
---|
166 | theta=60, phi=60, |
---|
167 | radius_pd=.2, radius_pd_n=9, |
---|
168 | length_pd=.2, length_pd_n=10, |
---|
169 | theta_pd=10, theta_pd_n=5, |
---|
170 | phi_pd=10, phi_pd_n=5) |
---|
171 | |
---|
172 | qx, 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 |
---|
174 | tests = [[{}, 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 [{'theta':10.0, 'phi':10.0}, (qx, qy), 0.03514647218513852], |
---|
180 | # [{'theta':10.0, 'phi':10.0}, [(qx, qy)], [0.03514647218513852]], |
---|
181 | ] |
---|
182 | del qx, qy # not necessary to delete, but cleaner |
---|
183 | # ADDED by: RKH ON: 18Mar2016 renamed sld's etc |
---|