.. _flexible-cylinder-elliptical:
flexible_cylinder_elliptical
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Flexible cylinder wth an elliptical cross section and a uniform scattering length density.
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Parameter Description Units Default value
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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
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
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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.
The form factor is normalized by the particle volume such that
.. math::
P(q) = \text{scale} \left/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 calculated in a similar way to that for the flexible_cylinder model
from the reference given 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 WRC. 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 $q0 = max(a3/sqrt(RgSquare),3)$ instead of
$max(a3*b/sqrt(RgSquare),3)$
- 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 (parameter 5) must be
greater than one.** Simple constraints should be applied during curve fitting to
maintain this inequality.
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
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.
**No inter-cylinder interference effects are included in this calculation.**
.. figure:: img/flexible_cylinder_elliptical_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