# 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 |
0.001 |

thickness |
sheet thickness |
Å |
33 |

Nlayers |
Number of layers |
None |
20 |

d_spacing |
lamellar spacing of paracrystal stack |
Å |
250 |

sigma_d |
Sigma (polydispersity) of the lamellar spacing |
Å |
0 |

sld |
layer scattering length density |
10 |
1 |

sld_solvent |
Solvent scattering length density |
10 |
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

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).

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

where

and

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

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

**Source**

`lamellar_stack_paracrystal.py`

\(\ \star\ \) `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:**