1 | r""" |
---|
2 | Definition |
---|
3 | ---------- |
---|
4 | |
---|
5 | Calculates the scattering from a barbell-shaped cylinder. Like |
---|
6 | :ref:`capped-cylinder`, this is a sphereocylinder with spherical end |
---|
7 | caps that have a radius larger than that of the cylinder, but with the center |
---|
8 | of the end cap radius lying outside of the cylinder. See the diagram for |
---|
9 | the details of the geometry and restrictions on parameter values. |
---|
10 | |
---|
11 | .. figure:: img/barbell_geometry.jpg |
---|
12 | |
---|
13 | Barbell geometry, where $r$ is *radius*, $R$ is *radius_bell* and |
---|
14 | $L$ is *length*. Since the end cap radius $R \geq r$ and by definition |
---|
15 | for this geometry $h < 0$, $h$ is then defined by $r$ and $R$ as |
---|
16 | $h = - \sqrt{R^2 - r^2}$ |
---|
17 | |
---|
18 | The scattered intensity $I(q)$ is calculated as |
---|
19 | |
---|
20 | .. math:: |
---|
21 | |
---|
22 | I(q) = \frac{\Delta \rho^2}{V} \left<A^2(q,\alpha).sin(\alpha)\right> |
---|
23 | |
---|
24 | where the amplitude $A(q,\alpha)$ with the rod axis at angle $\alpha$ to $q$ is given as |
---|
25 | |
---|
26 | .. math:: |
---|
27 | |
---|
28 | A(q) =&\ \pi r^2L |
---|
29 | \frac{\sin\left(\tfrac12 qL\cos\alpha\right)} |
---|
30 | {\tfrac12 qL\cos\alpha} |
---|
31 | \frac{2 J_1(qr\sin\alpha)}{qr\sin\alpha} \\ |
---|
32 | &\ + 4 \pi R^3 \int_{-h/R}^1 dt |
---|
33 | \cos\left[ q\cos\alpha |
---|
34 | \left(Rt + h + {\tfrac12} L\right)\right] |
---|
35 | \times (1-t^2) |
---|
36 | \frac{J_1\left[qR\sin\alpha \left(1-t^2\right)^{1/2}\right]} |
---|
37 | {qR\sin\alpha \left(1-t^2\right)^{1/2}} |
---|
38 | |
---|
39 | The $\left<\ldots\right>$ brackets denote an average of the structure over |
---|
40 | all orientations. $\left<A^2(q,\alpha)\right>$ is then the form factor, $P(q)$. |
---|
41 | The scale factor is equivalent to the volume fraction of cylinders, each of |
---|
42 | volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length |
---|
43 | densities of the cylinder and the surrounding solvent. |
---|
44 | |
---|
45 | The volume of the barbell is |
---|
46 | |
---|
47 | .. math:: |
---|
48 | |
---|
49 | V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right) |
---|
50 | |
---|
51 | |
---|
52 | and its radius of gyration is |
---|
53 | |
---|
54 | .. math:: |
---|
55 | |
---|
56 | R_g^2 =&\ \left[ \tfrac{12}{5}R^5 |
---|
57 | + R^4\left(6h+\tfrac32 L\right) |
---|
58 | + R^2\left(4h^2 + L^2 + 4Lh\right) |
---|
59 | + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\ |
---|
60 | &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3 |
---|
61 | + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right] |
---|
62 | \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1} |
---|
63 | |
---|
64 | .. note:: |
---|
65 | The requirement that $R \geq r$ is not enforced in the model! It is |
---|
66 | up to you to restrict this during analysis. |
---|
67 | |
---|
68 | The 2D scattering intensity is calculated similar to the 2D cylinder model. |
---|
69 | |
---|
70 | .. figure:: img/cylinder_angle_definition.png |
---|
71 | |
---|
72 | Definition of the angles for oriented 2D barbells. |
---|
73 | |
---|
74 | |
---|
75 | References |
---|
76 | ---------- |
---|
77 | |
---|
78 | .. [#] H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230 |
---|
79 | .. [#] H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda |
---|
80 | and errata) |
---|
81 | L. Onsager, Ann. New York Acad. Sci. 51, 627-659 (1949). |
---|
82 | |
---|
83 | Authorship and Verification |
---|
84 | ---------------------------- |
---|
85 | |
---|
86 | * **Author:** NIST IGOR/DANSE **Date:** pre 2010 |
---|
87 | * **Last Modified by:** Paul Butler **Date:** March 20, 2016 |
---|
88 | * **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017 |
---|
89 | """ |
---|
90 | |
---|
91 | import numpy as np |
---|
92 | from numpy import inf, sin, cos, pi |
---|
93 | |
---|
94 | name = "barbell" |
---|
95 | title = "Cylinder with spherical end caps" |
---|
96 | description = """ |
---|
97 | Calculates the scattering from a barbell-shaped cylinder. |
---|
98 | That is a sphereocylinder with spherical end caps that have a radius larger |
---|
99 | than that of the cylinder and the center of the end cap radius lies outside |
---|
100 | of the cylinder. |
---|
101 | Note: As the length of cylinder(bar) -->0,it becomes a dumbbell. And when |
---|
102 | rad_bar = rad_bell, it is a spherocylinder. |
---|
103 | It must be that rad_bar <(=) rad_bell. |
---|
104 | """ |
---|
105 | category = "shape:cylinder" |
---|
106 | # pylint: disable=bad-whitespace, line-too-long |
---|
107 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
108 | parameters = [["sld", "1e-6/Ang^2", 4, [-inf, inf], "sld", "Barbell scattering length density"], |
---|
109 | ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld", "Solvent scattering length density"], |
---|
110 | ["radius_bell", "Ang", 40, [0, inf], "volume", "Spherical bell radius"], |
---|
111 | ["radius", "Ang", 20, [0, inf], "volume", "Cylindrical bar radius"], |
---|
112 | ["length", "Ang", 400, [0, inf], "volume", "Cylinder bar length"], |
---|
113 | ["theta", "degrees", 60, [-360, 360], "orientation", "Barbell axis to beam angle"], |
---|
114 | ["phi", "degrees", 60, [-360, 360], "orientation", "Rotation about beam"], |
---|
115 | ] |
---|
116 | # pylint: enable=bad-whitespace, line-too-long |
---|
117 | |
---|
118 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "barbell.c"] |
---|
119 | have_Fq = True |
---|
120 | effective_radius_type = [ |
---|
121 | "equivalent cylinder excluded volume","equivalent volume sphere", "radius", "half length", "half total length", |
---|
122 | ] |
---|
123 | |
---|
124 | def random(): |
---|
125 | # TODO: increase volume range once problem with bell radius is fixed |
---|
126 | # The issue is that bell radii of more than about 200 fail at high q |
---|
127 | volume = 10**np.random.uniform(7, 9) |
---|
128 | bar_volume = 10**np.random.uniform(-4, -1)*volume |
---|
129 | bell_volume = volume - bar_volume |
---|
130 | bell_radius = (bell_volume/6)**0.3333 # approximate |
---|
131 | min_bar = bar_volume/np.pi/bell_radius**2 |
---|
132 | bar_length = 10**np.random.uniform(0, 3)*min_bar |
---|
133 | bar_radius = np.sqrt(bar_volume/bar_length/np.pi) |
---|
134 | if bar_radius > bell_radius: |
---|
135 | bell_radius, bar_radius = bar_radius, bell_radius |
---|
136 | pars = dict( |
---|
137 | #background=0, |
---|
138 | radius_bell=bell_radius, |
---|
139 | radius=bar_radius, |
---|
140 | length=bar_length, |
---|
141 | ) |
---|
142 | return pars |
---|
143 | |
---|
144 | # parameters for demo |
---|
145 | demo = dict(scale=1, background=0, |
---|
146 | sld=6, sld_solvent=1, |
---|
147 | radius_bell=40, radius=20, length=400, |
---|
148 | theta=60, phi=60, |
---|
149 | radius_pd=.2, radius_pd_n=5, |
---|
150 | length_pd=.2, length_pd_n=5, |
---|
151 | theta_pd=15, theta_pd_n=0, |
---|
152 | phi_pd=15, phi_pd_n=0, |
---|
153 | ) |
---|
154 | q = 0.1 |
---|
155 | # april 6 2017, rkh add unit tests, NOT compared with any other calc method, assume correct! |
---|
156 | qx = q*cos(pi/6.0) |
---|
157 | qy = q*sin(pi/6.0) |
---|
158 | tests = [ |
---|
159 | [{}, 0.075, 25.5691260532], |
---|
160 | [{'theta':80., 'phi':10.}, (qx, qy), 3.04233067789], |
---|
161 | ] |
---|
162 | del qx, qy # not necessary to delete, but cleaner |
---|