.. _lamellar-stack-paracrystal:

lamellar_stack_paracrystal
=======================================================

Random lamellar sheet with paracrystal 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|                   33
Nlayers     Number of layers                               None                    20
d_spacing   lamellar spacing of paracrystal stack          |Ang|                  250
sigma_d     Sigma (polydispersity) of the lamellar spacing |Ang|                    0
sld         layer scattering length density                |1e-6Ang^-2|             1
sld_solvent Solvent scattering length density              |1e-6Ang^-2|          6.34
=========== ============================================== ============ =============

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


This model calculates the scattering from a stack of repeating lamellar
structures. The stacks of lamellae (infinite in lateral dimension) are
treated as a paracrystal to account for the repeating spacing. The repeat
distance is further characterized by a Gaussian polydispersity. **This model
can be used for large multilamellar vesicles.**

**Definition**

In the equations below,

- *scale* is used instead of the mass per area of the bilayer $\Gamma_m$
  (this corresponds to the volume fraction of the material in the bilayer,
  *not* the total excluded volume of the paracrystal),

- *sld* $-$ *sld_solvent* is the contrast $\Delta \rho$,

- *thickness* is the layer thickness $t$,

- *Nlayers* is the number of layers $N$,

- *d_spacing* is the average distance between adjacent layers
  $\langle D \rangle$, and

- *sigma_d* is the relative standard deviation of the Gaussian
  layer distance distribution $\sigma_D / \langle D \rangle$.


The scattering intensity $I(q)$ is calculated as

.. math::

    I(q) = 2\pi\Delta\rho^2\Gamma_m\frac{P_\text{bil}(q)}{q^2} Z_N(q)

The form factor of the bilayer is approximated as the cross section of an
infinite, planar bilayer of thickness $t$ (compare the equations for the
lamellar model).

.. math::

    P_\text{bil}(q) = \left(\frac{\sin(qt/2)}{qt/2}\right)^2

$Z_N(q)$ describes the interference effects for aggregates
consisting of more than one bilayer. The equations used are (3-5)
from the Bergstrom reference:

.. math::


    Z_N(q) = \frac{1 - w^2}{1 + w^2 - 2w \cos(q \langle D \rangle)}
        + x_N S_N + (1 - x_N) S_{N+1}

where

.. math::

    S_N(q) = \frac{a_N}{N[1 + w^2 - 2 w \cos(q \langle D \rangle)]^2}

and

.. math::

    a_N &= 4w^2 - 2(w^3 + w) \cos(q \langle D \rangle) \\
        &\quad - 4w^{N+2}\cos(Nq \langle D \rangle)
        + 2 w^{N+3}\cos[(N-1)q \langle D \rangle]
        + 2w^{N+1}\cos[(N+1)q \langle D \rangle]

for the layer spacing distribution $w = \exp(-\sigma_D^2 q^2/2)$.

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

.. math::

    N_L = x_N N + (1 - x_N)(N+1)

The 2D scattering intensity is the same as 1D, regardless of the orientation
of the $q$ vector which is defined as

.. math::

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



.. figure:: img/lamellar_stack_paracrystal_autogenfig.png

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


**Source**

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

**Reference**

#. M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf,
   *J. Phys. Chem. B*, 103 (1999) 9888-9897

**Authorship and Verification**

* **Author:**
* **Last Modified by:**
* **Last Reviewed by:** Oliver Hammond, **Date:**  January 9, 2025

