source: sasmodels/sasmodels/models/barbell.py @ b0c4271

r"""
Definition
----------

Calculates the scattering from a barbell-shaped cylinder.  Like
:ref:`capped-cylinder`, this is a sphereocylinder 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 $h < 0$, $h$ 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)\right>

where the amplitude $A(q)$ is given as

.. math::

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

The $\left<\ldots\right>$ brackets denote an average of the structure over
all orientations. $\left<A^2(q)\right>$ 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^5
        + R^4\left(6h+\tfrac32 L\right)
        + R^2\left(4h^2 + L^2 + 4Lh\right)
        + R^2\left(3Lh^2 + \tfrac32 L^2h\right) \right. \\
        &\ \left. + \tfrac25 h^5 - \tfrac12 Lh^4 - \tfrac12 L^2h^3
        + \tfrac14 L^3r^2 + \tfrac32 Lr^4 \right]
        \left( 4R^3 6R^2h - 2h^3 + 3r^2L \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.jpg

    Definition of the angles for oriented 2D barbells.

.. figure:: img/cylinder_angle_projection.jpg
    :width: 600px

    Examples of the angles for oriented pp against the detector plane.

References
----------

.. [#] H Kaya, *J. Appl. Cryst.*, 37 (2004) 37 223-230
.. [#] H Kaya and N R deSouza, *J. Appl. Cryst.*, 37 (2004) 508-509 (addenda
   and errata)

Authorship and Verification
----------------------------

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Butler **Date:** March 20, 2016
* **Last Reviewed by:** Paul Butler **Date:** March 20, 2016
"""
from numpy import inf

name = "barbell"
title = "Cylinder with spherical end caps"
description = """
    Calculates the scattering from a barbell-shaped cylinder.
    That is a sphereocylinder with spherical end caps that have a radius larger
    than that of the cylinder and the center of the end cap radius lies outside
    of the cylinder.
    Note: As the length of cylinder(bar) -->0,it becomes a dumbbell. And when
    rad_bar = rad_bell, it is a spherocylinder.
    It must be that rad_bar <(=) rad_bell.
"""
category = "shape:cylinder"
# pylint: disable=bad-whitespace, line-too-long
#             ["name", "units", default, [lower, upper], "type","description"],
parameters = [["sld",         "1e-6/Ang^2",   4, [-inf, inf], "sld",         "Barbell scattering length density"],
              ["sld_solvent", "1e-6/Ang^2",   1, [-inf, inf], "sld",         "Solvent scattering length density"],
              ["radius_bell", "Ang",         40, [0, inf],    "volume",      "Spherical bell radius"],
              ["radius",      "Ang",         20, [0, inf],    "volume",      "Cylindrical bar radius"],
              ["length",      "Ang",        400, [0, inf],    "volume",      "Cylinder bar length"],
              ["theta",       "degrees",     60, [-inf, inf], "orientation", "In plane angle"],
              ["phi",         "degrees",     60, [-inf, inf], "orientation", "Out of plane angle"],
             ]
# pylint: enable=bad-whitespace, line-too-long

source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "barbell.c"]

# parameters for demo
demo = dict(scale=1, background=0,
            sld=6, sld_solvent=1,
            radius_bell=40, radius=20, length=400,
            theta=60, phi=60,
            radius_pd=.2, radius_pd_n=5,
            length_pd=.2, length_pd_n=5,
            theta_pd=15, theta_pd_n=0,
            phi_pd=15, phi_pd_n=0,
           )