# flexible_cylinder

Flexible cylinder where the form factor is normalized by the volume of the cylinder.

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 |

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 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.**

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

**Definitions**

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.

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

Our model uses the form factor calculations in reference [1] as implemented in a c-library provided by the NIST Center for Neutron Research (Kline, 2006). This states:

‘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.

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.

**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. In
particular, the cylinder form factor used as the limiting case for long
narrow rods will not be exactly correct for short and/or wide rods.**

**Source**

`flexible_cylinder.py`

\(\ \star\ \) `flexible_cylinder.c`

\(\ \star\ \) `wrc_cyl.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:****Last Reviewed by:**Steve King**Date:**March 6, 2020