# lamellar_hg_stack_caille

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

Parameter | Description | Units | Default value
--- | --- | --- | ---
scale | Scale factor or Volume fraction | 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 | 10-6Å-2 | 0.4
sld_head | Head scattering length density | 10-6Å-2 | 2
sld_solvent | Solvent scattering length density | 10-6Å-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

\[I(q) = 2 \pi \frac{P(q)S(q)}{q^2\delta }\]

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

\[P(q) = \frac{4}{q^2}\big\{ \Delta\rho_H \left[\sin[q(\delta_H + \delta_T)] - \sin(q\delta_T)\right] + \Delta\rho_T\sin(q\delta_T)\big\}^2\]

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

\[S(q) = 1 + 2 \sum_1^{N-1}\left(1-\frac{n}{N}\right) \cos(qdn)\exp\left(-\frac{2q^2d^2\alpha(n)}{2}\right)\]

where

\begin{align*} \alpha(n) &= \frac{\eta_{cp}}{4\pi^2} \left(\ln(\pi n)+\gamma_E\right) && \\ \gamma_E &= 0.5772156649 && \text{Euler's constant} \\ \eta_{cp} &= \frac{q_o^2k_B T}{8\pi\sqrt{K\overline{B}}} && \text{Caille constant} \end{align*}

\(\delta_T\) 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

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

Fig. 48 1D plot corresponding to the default parameters of the model.

Source

`lamellar_hg_stack_caille.py` \(\ \star\ \) `lamellar_hg_stack_caille.c`

References

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

2. J Berghausen, J Zipfel, P Lindner, W Richtering, J. Phys. Chem. B, 105, (2001) 11081-11088

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