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	10-6Å-2	6.3
sld_solvent	Solvent scattering length density	10-6Å-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

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

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

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

Fig. 43 1D plot corresponding to the default parameters of the model.

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