.. _lamellar-stack-caille:

lamellar_stack_caille
=======================================================

Random lamellar sheet with Caille structure factor

================ ================================= ============ =============
Parameter        Description                       Units        Default value
================ ================================= ============ =============
scale            Scale factor or Volume fraction   None                     1
background       Source background                 |cm^-1|              0.001
thickness        sheet thickness                   |Ang|                   30
Nlayers          Number of layers                  None                    20
d_spacing        lamellar d-spacing of Caille S(Q) |Ang|                  400
Caille_parameter Caille parameter                  |Ang^-2|               0.1
sld              layer scattering length density   |1e-6Ang^-2|           6.3
sld_solvent      Solvent scattering length density |1e-6Ang^-2|             1
================ ================================= ============ =============

The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.


This model provides the scattering intensity, $I(q) = P(q) S(q)$, for a
lamellar phase where a random distribution in solution are assumed.
Here a Caille $S(q)$ is used for the lamellar stacks.

**Definition**

The scattering intensity $I(q)$ is

.. math::

    I(q) = 2\pi \frac{P(q)S(q)}{q^2\delta }

The form factor is

.. math::

    P(q) = \frac{2\Delta\rho^2}{q^2}\left(1-\cos q\delta \right)

and the structure factor is

.. math::

    S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right)
           \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right)

where

.. math::
    :nowrap:

    \begin{align*}
    \alpha(n) &= \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right)
              && \\
    \gamma_E  &= 0.5772156649
              && \text{Euler's constant} \\
    \eta_{cp} &= \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}}
              && \text{Caille constant}
    \end{align*}

Here $d$ = (repeat) d_spacing, $\delta$ = bilayer thickness,
the contrast $\Delta\rho$ = SLD(headgroup) - SLD(solvent),
$K$ = smectic bending elasticity, $B$ = compression modulus, and
$N$ = number of lamellar plates (*n_plates*).

NB: **When the Caille parameter is greater than approximately 0.8 to 1.0, the
assumptions of the model are incorrect.** And due to a complication of the
model function, users are responsible for making sure that all the assumptions
are handled accurately (see the original reference below for more details).

Non-integer numbers of stacks are calculated as a linear combination of
results for the next lower and higher values.

The 2D scattering intensity is calculated in the same way as 1D, where the
$q$ vector is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}




.. figure:: img/lamellar_stack_caille_autogenfig.png

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


**Source**

:download:`lamellar_stack_caille.py <src/lamellar_stack_caille.py>`
$\ \star\ $ :download:`lamellar_stack_caille.c <src/lamellar_stack_caille.c>`

**References**

#. F Nallet, R Laversanne, and D Roux, *J. Phys. II France*, 3, (1993) 487-502
#. J Berghausen, J Zipfel, P Lindner, W Richtering,
   *J. Phys. Chem. B*, 105, (2001) 11081-11088

**Authorship and Verification**

* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**