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## pringle

The Pringle model provides the form factor, \(P(q)\), for a ‘pringle’ or ‘saddle-shaped’ disc that is bent in two directions.

Parameter | Description | Units | Default value
--- | --- | --- | ---
scale | Scale factor or Volume fraction | None | 1
background | Source background | cm-1 | 0.001
radius | Pringle radius | Å | 60
thickness | Thickness of pringle | Å | 10
alpha | Curvature parameter alpha | None | 0.001
beta | Curvature paramter beta | None | 0.02
sld | Pringle sld | 10-6Å-2 | 1
sld_solvent | Solvent sld | 10-6Å-2 | 6.3

The returned value is scaled to units of cm-1 sr-1, absolute scale.

Definition

The form factor for this bent disc is essentially that of a hyperbolic paraboloid and calculated as

\[P(q) = (\Delta \rho )^2 V \int^{\pi/2}_0 d\psi \sin{\psi} sinc^2 \left( \frac{qd\cos{\psi}}{2} \right) \left[ \left( S^2_0+C^2_0\right) + 2\sum_{n=1}^{\infty} \left( S^2_n+C^2_n\right) \right]\]

where

\[C_n = \frac{1}{r^2}\int^{R}_{0} r dr\cos(qr^2\alpha \cos{\psi}) J_n\left( qr^2\beta \cos{\psi}\right) J_{2n}\left( qr \sin{\psi}\right)\]
\[S_n = \frac{1}{r^2}\int^{R}_{0} r dr\sin(qr^2\alpha \cos{\psi}) J_n\left( qr^2\beta \cos{\psi}\right) J_{2n}\left( qr \sin{\psi}\right)\]

and \(\Delta\rho\text{ is }\rho_{pringle}-\rho_{solvent}\), \(V\) is the volume of the disc, \(\psi\) is the angle between the normal to the disc and the q vector, \(d\) and \(R\) are the “pringle” thickness and radius respectively, \(\alpha\) and \(\beta\) are the two curvature parameters, and \(J_n\) is the nth order Bessel function of the first kind.

Fig. 34 Figure 1: Schematic of model shape (Graphic from Matt Henderson, matt@matthen.com)

Fig. 35 Figure 2: 1D plot corresponding to the default parameters of the model.

Source

`pringle.py` \(\ \star\ \) `pringle.c` \(\ \star\ \) `adaptive.c` \(\ \star\ \) `sas_JN.c` \(\ \star\ \) `sas_J1.c` \(\ \star\ \) `sas_J0.c` \(\ \star\ \) `polevl.c`

Reference

1. Karen Edler, Universtiy of Bath, Private Communication. 2012. Derivation by Stefan Alexandru Rautu.

2. 12. Onsager, Ann. New York Acad. Sci., 51 (1949) 627-659

Authorship and Verification

- Author: Andrew Jackson Date: 2008

- Last Modified by: Wojciech Wpotrzebowski Date: March 20, 2016

- Last Reviewed by: Andrew Jackson Date: September 26, 2016