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1.. _barbell:
2
3Barbell
4=======================================================
5
6Cylinder with spherical end caps
7
8=========== ================================= ============ =============
9Parameter   Description                       Units        Default value
10=========== ================================= ============ =============
11scale       Source intensity                  None                     1
12background  Source background                 |cm^-1|                  0
13sld         Barbell scattering length density |1e-6Ang^-2|             4
14solvent_sld Solvent scattering length density |1e-6Ang^-2|             1
15bell_radius Spherical bell radius             |Ang|                   40
16radius      Cylindrical bar radius            |Ang|                   20
17length      Cylinder bar length               |Ang|                  400
18theta       In plane angle                    degree                  60
19phi         Out of plane angle                degree                  60
20=========== ================================= ============ =============
21
22The returned value is scaled to units of |cm^-1|.
23
24
25
26Calculates the scattering from a barbell-shaped cylinder (This model simply
27becomes the DumBellModel when the length of the cylinder, *L*, is set to zero).
28That is, a sphereocylinder with spherical end caps that have a radius larger
29than that of the cylinder and the center of the end cap radius lies outside
30of the cylinder. All dimensions of the BarBell are considered to be
31monodisperse. See the diagram for the details of the geometry and restrictions
32on parameter values.
33
34Definition
35----------
36
37The returned value is scaled to units of |cm^-1|\ |sr^-1|, absolute scale.
38
39The barbell geometry is defined as
40
41.. image:: img/barbell_geometry.jpg
42
43where *r* is the radius of the cylinder. All other parameters are as defined
44in the diagram.
45
46Since the end cap radius
47*R* >= *r* and by definition for this geometry *h* < 0, *h* is then
48defined by *r* and *R* as
49
50*h* = -1 \* sqrt(*R*\ :sup:`2` - *r*\ :sup:`2`)
51
52The scattered intensity *I(q)* is calculated as
53
54.. math::
55
56    I(Q) = \frac{(\Delta \rho)^2}{V} \left< A^2(Q)\right>
57
58where the amplitude *A(q)* is given as
59
60.. math::
61
62    A(Q) =&\ \pi r^2L
63        {\sin\left(\tfrac12 QL\cos\theta\right)
64            \over \tfrac12 QL\cos\theta}
65        {2 J_1(Qr\sin\theta) \over Qr\sin\theta} \\
66        &\ + 4 \pi R^3 \int_{-h/R}^1 dt
67        \cos\left[ Q\cos\theta
68            \left(Rt + h + {\tfrac12} L\right)\right]
69        \times (1-t^2)
70        {J_1\left[QR\sin\theta \left(1-t^2\right)^{1/2}\right]
71             \over QR\sin\theta \left(1-t^2\right)^{1/2}}
72
73The < > brackets denote an average of the structure over all orientations.
74<*A* :sup:`2`\ *(q)*> is then the form factor, *P(q)*. The scale factor is
75equivalent to the volume fraction of cylinders, each of volume, *V*. Contrast
76is the difference of scattering length densities of the cylinder and the
77surrounding solvent.
78
79The volume of the barbell is
80
81.. math::
82
83    V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right)
84
85
86and its radius-of-gyration is
87
88.. math::
89
90    R_g^2 =&\ \left[ \tfrac{12}{5}R^5
91        + R^4\left(6h+\tfrac32 L\right)
92        + R^2\left(4h^2 + L^2 + 4Lh\right)
93        + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\
94        &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3
95        + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right]
96        \left( 4R^3 6R^2h - 2h^3 + 3r^2L \right)^{-1}
97
98**The requirement that** *R* >= *r* **is not enforced in the model!** It is
99up to you to restrict this during analysis.
100
101This example dataset is produced by running the Macro PlotBarbell(),
102using 200 data points, *qmin* = 0.001 |Ang^-1|, *qmax* = 0.7 |Ang^-1|,
103*sld* = 4e-6 |Ang^-2| and the default model values.
104
105.. image:: img/barbell_1d.jpg
106
107*Figure. 1D plot using the default values (w/256 data point).*
108
109For 2D data: The 2D scattering intensity is calculated similar to the 2D
110cylinder model. For example, for |theta| = 45 deg and |phi| = 0 deg with
111default values for other parameters
112
113.. image:: img/barbell_2d.jpg
114
115*Figure. 2D plot (w/(256X265) data points).*
116
117.. image:: img/orientation.jpg
118
119Figure. Definition of the angles for oriented 2D barbells.
120
121.. image:: img/orientation2.jpg
122
123*Figure. Examples of the angles for oriented pp against the detector plane.*
124
125REFERENCE
126---------
127
128H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230
129
130H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda and errata)
131
132
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