# lamellar_stack_caille

Random lamellar sheet with Caille 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 |
30 |

Nlayers |
Number of layers |
None |
20 |

d_spacing |
lamellar d-spacing of Caille S(Q) |
Å |
400 |

Caille_parameter |
Caille parameter |
Å |
0.1 |

sld |
layer scattering length density |
10 |
6.3 |

sld_solvent |
Solvent scattering length density |
10 |
1 |

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

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.

**Definition**

The scattering intensity \(I(q)\) is

The form factor is

and the structure factor is

where

Here \(d\) = (repeat) d_spacing, \(\delta\) = bilayer thickness,
the contrast \(\Delta\rho\) = SLD(headgroup) - SLD(solvent),
\(K\) = smectic bending elasticity, \(B\) = compression modulus, and
\(N\) = number of lamellar plates (*n_plates*).

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

**Source**

`lamellar_stack_caille.py`

\(\ \star\ \) `lamellar_stack_caille.c`

**References**

F Nallet, R Laversanne, and D Roux,

*J. Phys. II France*, 3, (1993) 487-502J 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:**