.. _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 <src/pringle.py>`
$\ \star\ $ :download:`pringle.c <src/pringle.c>`
$\ \star\ $ :download:`gauss76.c <src/gauss76.c>`
$\ \star\ $ :download:`sas_JN.c <src/sas_JN.c>`
$\ \star\ $ :download:`sas_J1.c <src/sas_J1.c>`
$\ \star\ $ :download:`sas_J0.c <src/sas_J0.c>`
$\ \star\ $ :download:`polevl.c <src/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

