.. _flexible-cylinder-elliptical:

flexible_cylinder_elliptical
=======================================================

Flexible cylinder wth an elliptical cross section and a uniform scattering length density.

=========== ===================================== ============ =============
Parameter   Description                           Units        Default value
=========== ===================================== ============ =============
scale       Scale factor or Volume fraction       None                     1
background  Source background                     |cm^-1|              0.001
length      Length of the flexible cylinder       |Ang|                 1000
kuhn_length Kuhn length of the flexible cylinder  |Ang|                  100
radius      Radius of the flexible cylinder       |Ang|                   20
axis_ratio  Axis_ratio (major_radius/minor_radius None                   1.5
sld         Cylinder scattering length density    |1e-6Ang^-2|             1
sld_solvent Solvent scattering length density     |1e-6Ang^-2|           6.3
=========== ===================================== ============ =============

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


This model calculates the form factor for a flexible cylinder with an
elliptical cross section and a uniform scattering length density.
The non-negligible diameter of the cylinder is included by accounting
for excluded volume interactions within the walk of a single cylinder.
**Inter-cylinder interactions are NOT provided for.**

The form factor is normalized by the particle volume such that

.. math::

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

where the averaging $\left<\ldots\right>$ is over all possible orientations
of the flexible cylinder.

The 2D scattering intensity is the same as 1D, regardless of the orientation
of the q vector which is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}


**Definitions**

The function is calculated in a similar way to that for the
:ref:`flexible-cylinder` model in reference [1] below using the author's
"Method 3 With Excluded Volume".

The model is a parameterization of simulations of a discrete representation of
the worm-like chain model of Kratky and Porod applied in the pseudo-continuous
limit. See equations (13, 26-27) in the original reference for the details.

.. note::

    There are several typos in the original reference that have been
    corrected by Chen *et al* (WRC) [2]. Details of the corrections are in the
    reference below. Most notably

    - Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$

    - Equations (23) and (24) are incorrect; WRC has entered these into
      Mathematica and solved analytically. The results were then converted to
      code.

    - Equation (27) should be $q0 = max(a3/(Rg^2)^{1/2},3)$ instead of
      $max(a3*b(Rg^2)^{1/2},3)$

    - The scattering function is negative for a range of parameter values and
      q-values that are experimentally accessible. A correction function has
      been added to give the proper behavior.

.. figure:: img/flexible_cylinder_ex_geometry.jpg


The chain of contour length, $L$, (the total length) can be described as a
chain of some number of locally stiff segments of length $l_p$, the
persistence length (the length along the cylinder over which the flexible
cylinder can be considered a rigid rod). The Kuhn length $(b = 2*l_p)$ is
also used to describe the stiffness of a chain.

The cross section of the cylinder is elliptical, with minor radius $a$ . The
major radius is larger, so of course, **the axis_ratio must be greater than
one.** Simple constraints should be applied during curve fitting to maintain
this inequality.

In the parameters, the $sld$ and $sld\_solvent$ represent the SLD of the
chain/cylinder and solvent respectively. The *scale*, and the contrast are
both multiplicative factors in the model and are perfectly correlated. One or
both of these parameters must be held fixed during model fitting.

**This is a model with complex behaviour depending on the ratio of** $L/b$
**and the reader is strongly encouraged to read reference [1] before use.**


.. figure:: img/flexible_cylinder_elliptical_autogenfig.png

    1D plot corresponding to the default parameters of the model.


**Source**

:download:`flexible_cylinder_elliptical.py <src/flexible_cylinder_elliptical.py>`
$\ \star\ $ :download:`flexible_cylinder_elliptical.c <src/flexible_cylinder_elliptical.c>`
$\ \star\ $ :download:`wrc_cyl.c <src/wrc_cyl.c>`
$\ \star\ $ :download:`gauss76.c <src/gauss76.c>`
$\ \star\ $ :download:`sas_J1.c <src/sas_J1.c>`
$\ \star\ $ :download:`polevl.c <src/polevl.c>`

**References**

#. J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible
   polymers with and without excluded volume effects.*
   Macromolecules, 29 (1996) 7602-7612
#. W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar
   Interactions in the Fitting of SANS Data from Cationic Wormlike Micelles.*
   Langmuir, 22(15) 2006 6539-6548

**Authorship and Verification**

* **Author:**
* **Last Modified by:** Richard Heenan **Date:** December, 2016
* **Last Reviewed by:** Steve King **Date:** March 26, 2019