# lamellar_stack_caille

Random lamellar sheet with Caille structure factor

Parameter | Description | Units | Default value |
---|---|---|---|

scale | Source intensity | None | 1 |

background | Source background | cm^{-1} |
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^{-2} |
0.1 |

sld | layer scattering length density | 10^{-6}Å^{-2} |
6.3 |

sld_solvent | Solvent scattering length density | 10^{-6}Å^{-2} |
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

**References**

F Nallet, R Laversanne, and D Roux, J. Phys. II France, 3, (1993) 487-502

also in J. Phys. Chem. B, 105, (2001) 11081-11088