.. _pringle: 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 |Ang| 60 thickness Thickness of pringle |Ang| 10 alpha Curvature parameter alpha None 0.001 beta Curvature paramter beta None 0.02 sld Pringle sld |1e-6Ang^-2| 1 sld_solvent Solvent sld |1e-6Ang^-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 .. math:: 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 .. math:: 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) .. math:: 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\ :sup:`th` order Bessel function of the first kind. .. figure:: img/pringles_fig1.png Schematic of model shape (Graphic from Matt Henderson, matt@matthen.com) .. figure:: img/pringle_autogenfig.png 1D plot corresponding to the default parameters of the model. **Source** :download:`pringle.py ` $\ \star\ $ :download:`pringle.c ` $\ \star\ $ :download:`gauss76.c ` $\ \star\ $ :download:`sas_JN.c ` $\ \star\ $ :download:`sas_J1.c ` $\ \star\ $ :download:`sas_J0.c ` $\ \star\ $ :download:`polevl.c ` **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