.. _flexible-cylinder-elliptical: flexible_cylinder_elliptical ======================================================= Flexible cylinder wth an elliptical cross section and a uniform scattering length density. =========== ===================================== ============ ============= 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 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 =========== ===================================== ============ ============= 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