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	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	10-6Å-2	1
sld_solvent	Solvent scattering length density	10-6Å-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.

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

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

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

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.

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.

Fig. 29 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