.. _lamellar:

lamellar
=======================================================

Lyotropic lamellar phase with uniform SLD and random distribution

=========== ================================= ============ =============
Parameter   Description                       Units        Default value
=========== ================================= ============ =============
scale       Scale factor or Volume fraction   None                     1
background  Source background                 |cm^-1|              0.001
thickness   total layer thickness             |Ang|                   50
sld         Layer scattering length density   |1e-6Ang^-2|             1
sld_solvent Solvent scattering length density |1e-6Ang^-2|             6
=========== ================================= ============ =============

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


Polydispersity in the bilayer thickness can be applied from the GUI.

**Definition**

The scattering intensity $I(q)$ for dilute, randomly oriented,
"infinitely large" sheets or lamellae is

.. math::

    I(q) = \text{scale}\frac{2\pi P(q)}{q^2\delta} + \text{background}


The form factor is

.. math::

   P(q) = \frac{2\Delta\rho^2}{q^2}(1-\cos(q\delta))
        = \frac{4\Delta\rho^2}{q^2}\sin^2\left(\frac{q\delta}{2}\right)

where $\delta$ is the total layer thickness and $\Delta\rho$ is the
scattering length density difference.

This is the limiting form for a spherical shell of infinitely large radius.
Note that the division by $\delta$ means that $scale$ in sasview is the
volume fraction of sheet, $\phi = S\delta$ where $S$ is the area of sheet
per unit volume. $S$ is half the Porod surface area per unit volume of a
thicker layer (as that would include both faces of the sheet).

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_autogenfig.png

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


**Source**

:download:`lamellar.py <src/lamellar.py>`

**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

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