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.

../../_images/pringles_fig1.png

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

../../_images/pringle_autogenfig.png

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

Source

pringle.py \(\ \star\ \) pringle.c \(\ \star\ \) lib/gauss76.c \(\ \star\ \) lib/sas_JN.c \(\ \star\ \) lib/sas_J1.c \(\ \star\ \) lib/sas_J0.c \(\ \star\ \) lib/polevl.c

Reference

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