.. _cylinder:

cylinder
=======================================================

Right circular cylinder with uniform scattering length density.

=========== ================================== ============ =============
Parameter   Description                        Units        Default value
=========== ================================== ============ =============
scale       Scale factor or Volume fraction    None                     1
background  Source background                  |cm^-1|              0.001
sld         Cylinder scattering length density |1e-6Ang^-2|             4
sld_solvent Solvent scattering length density  |1e-6Ang^-2|             1
radius      Cylinder radius                    |Ang|                   20
length      Cylinder length                    |Ang|                  400
theta       cylinder 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.



For information about polarised and magnetic scattering, see
the :ref:`magnetism` documentation.

**Definition**

The output of the 2D scattering intensity function for oriented cylinders is
given by (Guinier, 1955)

.. math::

    I(q,\alpha) = \frac{\text{scale}}{V} F^2(q,\alpha) + \text{background}

where

.. math::

    F(q,\alpha) = 2 (\Delta \rho) V
           \frac{\sin \left(\tfrac12 qL\cos\alpha \right)}
                {\tfrac12 qL \cos \alpha}
           \frac{J_1 \left(q R \sin \alpha\right)}{q R \sin \alpha}

and $\alpha$ is the angle between the axis of the cylinder and $\vec q$,
$V =\pi R^2L$ is the volume of the cylinder, $L$ is the length of the cylinder,
$R$ is the radius of the cylinder, and $\Delta\rho$ (contrast) is the
scattering length density difference between the scatterer and the solvent.
$J_1$ is the first order Bessel function.

For randomly oriented particles:

.. math::

    P(q)=F^2(q)=\int_{0}^{\pi/2}{F^2(q,\alpha)\sin(\alpha)d\alpha}


The output of the 1D scattering intensity function for randomly oriented
cylinders is thus given by

.. math::

    I(q) = \frac{\text{scale}}{V}
        \int_0^{\pi/2} F^2(q,\alpha) \sin \alpha\ d\alpha + \text{background}


NB: The 2nd virial coefficient of the cylinder is calculated based on the
radius and length values, and used as the effective radius for $S(q)$
when $P(q) \cdot S(q)$ is applied.

For 2d scattering from oriented cylinders, we define the direction of the
axis of the cylinder using two angles $\theta$ (note this is not the same as
the scattering angle used in q) and $\phi$. Those angles are defined in
:numref:`cylinder-angle-definition` , for further details see
:ref:`orientation`.

.. _cylinder-angle-definition:

.. figure:: img/cylinder_angle_definition.png

    Angles $\theta$ and $\phi$ orient the cylinder relative to the beam line
    coordinates, where the beam is along the $z$ axis. Rotation $\theta$,
    initially in the $xz$ plane, is carried out first, then rotation $\phi$
    about the $z$ axis. Orientation distributions are described as rotations
    about two perpendicular axes $\delta_1$ and $\delta_2$ in the frame of
    the cylinder itself, which when $\theta = \phi = 0$ are parallel to the
    $Y$ and $X$ axes.

.. figure:: img/cylinder_angle_projection.png

    Examples for oriented cylinders.

The $\theta$ and $\phi$ parameters to orient the cylinder only appear in the
model when fitting 2d data.

**Validation**

Validation of the code was done by comparing the output of the 1D model
to the output of the software provided by the NIST (Kline, 2006).
The implementation of the intensity for fully oriented cylinders was done
by averaging over a uniform distribution of orientations using

.. math::

    P(q) = \int_0^{\pi/2} d\phi
        \int_0^\pi p(\theta) P_0(q,\theta) \sin \theta\ d\theta


where $p(\theta,\phi) = 1$ is the probability distribution for the orientation
and $P_0(q,\theta)$ is the scattering intensity for the fully oriented
system, and then comparing to the 1D result.


.. figure:: img/cylinder_autogenfig.png

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


**Source**

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

#.  J. Pedersen, *Adv. Colloid Interface Sci.*, 70 (1997) 171-210
#.  G. Fournet, *Bull. Soc. Fr. Mineral. Cristallogr.*, 74 (1951) 39-113
#.  L. Onsager, *Ann. New York Acad. Sci.*, 51 (1949) 627-659

**Authorship and Verification**

* **Author:**
* **Last Modified by:** Paul Butler (docs only) November 10, 2022
* **Last Reviewed by:**

