.. _flexible-cylinder:

flexible_cylinder
=======================================================

Flexible cylinder where the form factor is normalized by the volumeof the cylinder.

=========== ==================================== ============ =============
Parameter   Description                          Units        Default value
=========== ==================================== ============ =============
scale       Source intensity                     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
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 provides the form factor, $P(q)$, for a flexible cylinder
where the form factor is normalized by the volume of the cylinder.
**Inter-cylinder interactions are NOT provided for.**

.. math::

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

where the averaging $\left<\ldots\right>$ is applied only for the 1D
calculation

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**

.. figure:: img/flexible_cylinder_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 returned value is in units of $cm^{-1}$, on absolute scale.

In the parameters, the sld and sld\_solvent represent the SLD of the cylinder
and solvent respectively.

Our model uses the form factor calculations implemented in a c-library provided
by the NIST Center for Neutron Research (Kline, 2006).


From the reference:

    'Method 3 With Excluded Volume' is used.
    The model is a parametrization of simulations of a discrete representation
    of the worm-like chain model of Kratky and Porod applied in the
    pseudocontinuous limit.
    See equations (13,26-27) in the original reference for the details.


.. figure:: img/flexible_cylinder_autogenfig.png

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

**References**

J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible
polymers with and without excluded volume effects.* Macromolecules,
29 (1996) 7602-7612

Correction of the formula can be found in

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