Lamellar Stack Caille

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

$$ I(q) = 2\pi \frac{P(q)S(q)}{q^2\delta }
$$
The form factor is

$$ P(q) = \frac{2\Delta\rho^2}{q^2}\left(1-\cos q\delta \right)
$$
and the structure factor 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

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

$$ q = \sqrt{q_x^2 + q_y^2}
$$

References

#. F Nallet, R Laversanne, and D Roux, *J. Phys. II France*, 3, (1993) 487-502 #. J Berghausen, J Zipfel, P Lindner, W Richtering, *J. Phys. Chem. B*, 105, (2001) 11081-11088

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