# 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 |
0.001 |

length |
Length of the flexible cylinder |
Å |
1000 |

kuhn_length |
Kuhn length of the flexible cylinder |
Å |
100 |

radius |
Radius of the flexible cylinder |
Å |
20 |

axis_ratio |
Axis_ratio (major_radius/minor_radius |
None |
1.5 |

sld |
Cylinder scattering length density |
10 |
1 |

sld_solvent |
Solvent scattering length density |
10 |
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

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

**Definitions**

The function is calculated in a similar way to that for the 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.

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.**

**Source**

`flexible_cylinder_elliptical.py`

\(\ \star\ \) `flexible_cylinder_elliptical.c`

\(\ \star\ \) `wrc_cyl.c`

\(\ \star\ \) `gauss76.c`

\(\ \star\ \) `sas_J1.c`

\(\ \star\ \) `polevl.c`

**References**

J S Pedersen and P Schurtenberger.

*Scattering functions of semiflexible polymers with and without excluded volume effects.*Macromolecules, 29 (1996) 7602-7612W 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