.. _barbell:

barbell
=======================================================

Cylinder with spherical end caps

=========== ================================= ============ =============
Parameter   Description                       Units        Default value
=========== ================================= ============ =============
scale       Scale factor or Volume fraction   None                     1
background  Source background                 |cm^-1|              0.001
sld         Barbell scattering length density |1e-6Ang^-2|             4
sld_solvent Solvent scattering length density |1e-6Ang^-2|             1
radius_bell Spherical bell radius             |Ang|                   40
radius      Cylindrical bar radius            |Ang|                   20
length      Cylinder bar length               |Ang|                  400
theta       Barbell axis to beam angle        degree                  60
phi         Rotation about beam               degree                  60
=========== ================================= ============ =============

The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.


**Definition**

Calculates the scattering from a barbell-shaped cylinder.  Like
:ref:`capped-cylinder`, this is a spherocylinder with spherical end
caps that have a radius larger than that of the cylinder, but with the center
of the end cap radius lying outside of the cylinder. See the diagram for
the details of the geometry and restrictions on parameter values.

.. figure:: img/barbell_geometry.jpg

    Barbell geometry, where $r$ is *radius*, $R$ is *radius_bell* and
    $L$ is *length*. Since the end cap radius $R \geq r$ and by definition
    for this geometry $h \ge 0$, $h$ is then defined by $r$ and $R$ as
    $h = \sqrt{R^2 - r^2}$

The scattered intensity $I(q)$ is calculated as

.. math::

    I(q) = \frac{\Delta \rho^2}{V} \left<A^2(q,\alpha).sin(\alpha)\right>

where the amplitude $A(q,\alpha)$ with the rod axis at angle $\alpha$ to $q$
is given as

.. math::

    A(q) =&\ \pi r^2L
        \frac{\sin\left(\tfrac12 qL\cos\alpha\right)}
             {\tfrac12 qL\cos\alpha}
        \frac{2 J_1(qr\sin\alpha)}{qr\sin\alpha} \\
        &\ + 4 \pi R^3 \int_{-h/R}^1 dt
        \cos\left[ q\cos\alpha
            \left(Rt + h + {\tfrac12} L\right)\right]
        \times (1-t^2)
        \frac{J_1\left[qR\sin\alpha \left(1-t^2\right)^{1/2}\right]}
             {qR\sin\alpha \left(1-t^2\right)^{1/2}}

The $\left<\ldots\right>$ brackets denote an average of the structure over
all orientations. $\left<A^2(q,\alpha)\right>$ is then the form factor, $P(q)$.
The scale factor is equivalent to the volume fraction of cylinders, each of
volume, $V$. Contrast $\Delta\rho$ is the difference of scattering length
densities of the cylinder and the surrounding solvent.

The volume of the barbell is

.. math::

    V = \pi r_c^2 L + 2\pi\left(\tfrac23R^3 + R^2h-\tfrac13h^3\right)

and its radius of gyration is

.. math::

    R_g^2 =&\ \left[ \tfrac{12}{5}R^4
        + R^3\left(3L + \tfrac{18}{5} h\right)
        + R^2\left(L^2 + Lh + \tfrac25 h^2\right)
        + R\left(\tfrac14 L^3 + \tfrac12 L^2h - Lh^2\right) \right. \\
        &\ \left. + Lh^4 - \tfrac12 L^2h^3 - \tfrac14 L^3h + \tfrac25 h^4\right]
        \left( 4R^2 + 3LR + 2Rh - 3Lh - 2h^2\right)^{-1}

.. note::
    The requirement that $R \geq r$ is not enforced in the model! It is
    up to you to restrict this during analysis.

The 2D scattering intensity is calculated similar to the 2D cylinder model.

.. figure:: img/cylinder_angle_definition.png

    Definition of the angles for oriented 2D barbells.



.. figure:: img/barbell_autogenfig.png

    1D and 2D plots corresponding to the default parameters of the model.


**Source**

:download:`barbell.py <src/barbell.py>`
$\ \star\ $ :download:`barbell.c <src/barbell.c>`
$\ \star\ $ :download:`adaptive.c <src/adaptive.c>`
$\ \star\ $ :download:`sas_J1.c <src/sas_J1.c>`
$\ \star\ $ :download:`polevl.c <src/polevl.c>`

**References**

#. H Kaya, *J. Appl. Cryst.*, 37 (2004) 223-230

#. H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509
   (addenda and errata)

#. L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659

**Authorship and Verification**

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Butler **Date:** March 20, 2016
* **Last Reviewed by:** Richard Heenan **Date:** January 4, 2017

