.. _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       Source intensity          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.

**Reference**

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