.. _hollow-cylinder:

hollow_cylinder
=======================================================



=========== =============================== ============ =============
Parameter   Description                     Units        Default value
=========== =============================== ============ =============
scale       Scale factor or Volume fraction None                     1
background  Source background               |cm^-1|              0.001
radius      Cylinder core radius            |Ang|                   20
thickness   Cylinder wall thickness         |Ang|                   10
length      Cylinder total length           |Ang|                  400
sld         Cylinder sld                    |1e-6Ang^-2|           6.3
sld_solvent Solvent sld                     |1e-6Ang^-2|             1
theta       Cylinder axis to beam angle     degree                  90
phi         Rotation about beam             degree                   0
=========== =============================== ============ =============

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


**Definition**

This model provides the form factor, $P(q)$, for a monodisperse hollow right
angle circular cylinder (rigid tube) where the The inside and outside of the
hollow cylinder are assumed to have the same SLD and the form factor is thus
normalized by the volume of the tube (i.e. not by the total cylinder volume).

.. math::

    P(q) = \text{scale} \left<F^2\right>/V_\text{shell} + \text{background}

where the averaging $\left<\ldots\right>$ is applied only for the 1D
calculation. If Intensity is given on an absolute scale, the scale factor here
is the volume fraction of the shell.  This differs from
the :ref:`core-shell-cylinder` in that, in that case, scale is the volume
fraction of the entire cylinder (core+shell). The application might be for a
bilayer which wraps into a hollow tube and the volume fraction of material is
all in the shell, whereas the :ref:`core-shell-cylinder` model might be used for
a cylindrical micelle where the tails in the core have a different SLD than the
headgroups (in the shell) and the volume fraction of material comes fromm the
whole cyclinder.  NOTE: the hollow_cylinder represents a tube whereas the
core_shell_cylinder includes a shell layer covering the ends (end caps) as well.


The 1D scattering intensity is calculated in the following way (Guinier, 1955)

.. math::

    P(q)           &= (\text{scale})V_\text{shell}\Delta\rho^2
            \int_0^{1}\Psi^2
            \left[q_z, R_\text{outer}(1-x^2)^{1/2},
                       R_\text{core}(1-x^2)^{1/2}\right]
            \left[\frac{\sin(qHx)}{qHx}\right]^2 dx \\
    \Psi[q,y,z]    &= \frac{1}{1-\gamma^2}
            \left[ \Lambda(qy) - \gamma^2\Lambda(qz) \right] \\
    \Lambda(a)     &= 2 J_1(a) / a \\
    \gamma         &= R_\text{core} / R_\text{outer} \\
    V_\text{shell} &= \pi \left(R_\text{outer}^2 - R_\text{core}^2 \right)L \\
    J_1(x)         &= (\sin(x)-x\cdot \cos(x)) / x^2

where *scale* is a scale factor, $H = L/2$ and $J_1$ is the 1st order
Bessel function.

**NB**: The 2nd virial coefficient of the cylinder is calculated
based on the outer radius and full length, which give an the effective radius
for structure factor $S(q)$ when $P(q) \cdot S(q)$ is applied.

In the parameters,the *radius* is $R_\text{core}$ while *thickness*
is $R_\text{outer} - R_\text{core}$.

To provide easy access to the orientation of the core-shell cylinder, we define
the axis of the cylinder using two angles $\theta$ and $\phi$
(see :ref:`cylinder model <cylinder-angle-definition>`).


.. figure:: img/hollow_cylinder_autogenfig.png

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


**Source**

:download:`hollow_cylinder.py <src/hollow_cylinder.py>`
$\ \star\ $ :download:`hollow_cylinder.c <src/hollow_cylinder.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**

#. L A Feigin and D I Svergun, *Structure Analysis by Small-Angle X-Ray and
   Neutron Scattering*, Plenum Press, New York, (1987)
#. 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:** September 06, 2018
   (corrected VR calculation)
* **Last Reviewed by:** Paul Butler **Date:** September 06, 2018

