.. _lamellarCailleHG:

Lamellarcaillehg
=======================================================

Random lamellar sheet with Caille structure factor

================ ================================= ============ =============
Parameter        Description                       Units        Default value
================ ================================= ============ =============
scale            Source intensity                  None                     1
background       Source background                 |cm^-1|                  0
tail_length      Tail thickness                    |Ang|                   10
head_length      head thickness                    |Ang|                    2
Nlayers          Number of layers                  None                    30
spacing          d-spacing of Caille S(Q)          |Ang|                   40
Caille_parameter Caille parameter                  None                 0.001
sld              Tail scattering length density    |1e-6Ang^-2|           0.4
head_sld         Head scattering length density    |1e-6Ang^-2|             2
solvent_sld      Solvent scattering length density |1e-6Ang^-2|             6
================ ================================= ============ =============

The returned value is scaled to units of |cm^-1|.


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.

The scattering intensity $I(q)$ is

.. math::

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


The form factor $P(q)$ is

.. math::

        P(q) = \frac{4}{q^2}\big\{
        \Delta\rho_H \left[\sin[q(\delta_H + \delta_T)] - \sin(q\delta_T)\right]
            + \Delta\rho_T\sin(q\delta_T)\big\}^2

and the structure factor $S(q)$ 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::

    \begin{eqnarray}
    \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{eqnarray}


$\delta_T$ is the tail length (or *tail_length*), $\delta_H$ is the head
thickness (or *head_length*), $\Delta\rho_H$ is SLD(headgroup) - SLD(solvent),
and $\Delta\rho_T$ is SLD(tail) - SLD(headgroup). Here $d$ is (repeat) spacing,
$K$ is smectic bending elasticity, $B$ is compression modulus, and $N$ is the
number of lamellar plates (*Nlayers*).

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}

The returned value is in units of |cm^-1|, on absolute scale.

.. image:: img/lamellarCailleHG_1d.jpg

*Figure. 1D plot using the default values (w/6000 data point).*

Our model uses the form factor calculations implemented in a C library provided
by the NIST Center for Neutron Research (Kline, 2006).

REFERENCE

F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502

also in J. Phys. Chem. B, 105, (2001) 11081-11088