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
radius Pringle radius 60
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.


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

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


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

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. 29 Schematic of model shape (Graphic from Matt Henderson,


Fig. 30 1D plot corresponding to the default parameters of the model.


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

Author: Andrew Jackson on: 2008

Last Modified by: Wojciech Wpotrzebowski on: March 20, 2016

Last Reviewed by: Andrew Jackson on: September 26, 2016