# lamellar_hg_stack_caille

Random lamellar head/tail/tail/head sheet with Caille structure factor

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

scale | Source intensity | None | 1 |

background | Source background | cm^{-1} |
0.001 |

length_tail | Tail thickness | Å | 10 |

length_head | head thickness | Å | 2 |

Nlayers | Number of layers | None | 30 |

d_spacing | lamellar d-spacing of Caille S(Q) | Å | 40 |

Caille_parameter | Caille parameter | None | 0.001 |

sld | Tail scattering length density | 10^{-6}Å^{-2} |
0.4 |

sld_head | Head scattering length density | 10^{-6}Å^{-2} |
2 |

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

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.

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

The form factor \(P(q)\) is

and the structure factor \(S(q)\) is

where

\(\delta_T\) is the tail length (or *length_tail*), \(\delta_H\) is the head
thickness (or *length_head*), \(\Delta\rho_H\) is SLD(headgroup) - SLD(solvent),
and \(\Delta\rho_T\) is SLD(tail) - SLD(headgroup). Here \(d\) is (repeat) spacing,
\(K\) is smectic bending elasticity, \(B\) is compression modulus, and \(N\) is the
number of lamellar plates (*Nlayers*).

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.

Be aware that the computations may be very slow.

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