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

r"""
Definitions
-----------

Calculates the scattering from a cylinder with spherical section end-caps.
Like :ref:`barbell`, this is a sphereocylinder with end caps that have a
radius larger than that of the cylinder, but with the center of the end cap
radius lying within the cylinder. This model simply becomes a convex
lens when the length of the cylinder $L=0$. See the diagram for the details
of the geometry and restrictions on parameter values.

.. figure:: img/capped_cylinder_geometry.jpg

    Capped cylinder geometry, where $r$ is *radius*, $R$ is *bell_radius* 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 capped cylinder is (with $h$ as a positive value here)

.. math::

    V = \pi r_c^2 L + \tfrac{2\pi}{3}(R-h)^2(2R + h)


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 cylinders.

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

    Examples of the angles for oriented 2D cylinders against the detector plane.

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)

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

* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Paul Butler **Date:** September 30, 2016
* **Last Reviewed by:** Richard Heenan **Date:** March 19, 2016
"""
from numpy import inf

name = "capped_cylinder"
title = "Right circular cylinder with spherical end caps and uniform SLD"
description = """That is, a sphereocylinder
    with end caps that have a radius larger than
    that of the cylinder and the center of the
    end cap radius lies within the cylinder.
    Note: As the length of cylinder -->0,
    it becomes a Convex Lens.
    It must be that radius <(=) radius_cap.
    [Parameters];
    scale: volume fraction of spheres,
    background:incoherent background,
    radius: radius of the cylinder,
    length: length of the cylinder,
    radius_cap: radius of the semi-spherical cap,
    sld: SLD of the capped cylinder,
    sld_solvent: SLD of the solvent.
"""
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",    "Cylinder scattering length density"],
              ["sld_solvent", "1e-6/Ang^2", 1, [-inf, inf], "sld",    "Solvent scattering length density"],
              ["radius",      "Ang",       20, [0, inf],    "volume", "Cylinder radius"],

              # TODO: use an expression for cap radius with fixed bounds.
              # The current form requires cap radius R bigger than cylinder radius r.
              # Could instead use R/r in [1,inf], r/R in [0,1], or the angle between
              # cylinder and cap in [0,90].  The problem is similar for the barbell
              # model.  Propose r/R in [0,1] in both cases, with the model specifying
              # cylinder radius in the capped cylinder model and sphere radius in the
              # barbell model.  This leads to the natural value of zero for no cap
              # in the capped cylinder, and zero for no bar in the barbell model.  In
              # both models, one would be a pill.
              ["radius_cap", "Ang",     20, [0, inf],    "volume", "Cap radius"],
              ["length",     "Ang",    400, [0, inf],    "volume", "Cylinder 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", "capped_cylinder.c"]

demo = dict(scale=1, background=0,
            sld=6, sld_solvent=1,
            radius=260, radius_cap=290, length=290,
            theta=30, phi=15,
            radius_pd=.2, radius_pd_n=1,
            radius_cap_pd=.2, radius_cap_pd_n=1,
            length_pd=.2, length_pd_n=10,
            theta_pd=15, theta_pd_n=45,
            phi_pd=15, phi_pd_n=1)