# 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 Source intensity None 1
background Source background cm-1 0.001
thickness Thickness of pringle 10
alpha Curvature parameter alpha None 0.001
beta Curvature paramter beta None 0.02
sld Pringle sld 10-6-2 1
sld_solvent Solvent sld 10-6-2 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.

Reference

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

Author: Andrew Jackson on: 2008