.. _lamellar-hg:

lamellar_hg
=======================================================

Random lamellar phase with Head and Tail Groups

=========== ================================= ============ =============
Parameter   Description                       Units        Default value
=========== ================================= ============ =============
scale       Scale factor or Volume fraction   None                     1
background  Source background                 |cm^-1|              0.001
length_tail Tail thickness ( total = H+T+T+H) |Ang|                   15
length_head Head thickness                    |Ang|                   10
sld         Tail scattering length density    |1e-6Ang^-2|           0.4
sld_head    Head scattering length density    |1e-6Ang^-2|             3
sld_solvent Solvent scattering length density |1e-6Ang^-2|             6
=========== ================================= ============ =============

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


This model provides the scattering intensity, $I(q)$, for a lyotropic lamellar
phase where a random distribution in solution are assumed. The SLD of the head
region is taken to be different from the SLD of the tail region.

**Definition**

The scattering intensity $I(q)$ is

.. math::

   I(q) = 2\pi\frac{\text{scale}}{2(\delta_H + \delta_T)}  P(q) \frac{1}{q^2}

The form factor $P(q)$ is

.. math::

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

where $\delta_T$ is *length_tail*, $\delta_H$ is *length_head*,
$\Delta\rho_H$ is the head contrast (*sld_head* $-$ *sld_solvent*),
and $\Delta\rho_T$ is tail contrast (*sld* $-$ *sld_solvent*).

The total thickness of the lamellar sheet is
$a_H + \delta_T + \delta_T + \delta_H$. Note that in a non aqueous solvent
the chemical "head" group may be the "Tail region" and vice-versa.

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_hg_autogenfig.png

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


**Source**

:download:`lamellar_hg.py <src/lamellar_hg.py>`
$\ \star\ $ :download:`lamellar_hg.c <src/lamellar_hg.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:** S King and P Butler **Date** April 17, 2014

