Pringle

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 n`th` order Bessel function of the first kind.

Schematic of model shape (Graphic from Matt Henderson, matt@matthen.com)

Reference

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