Lamellar

Polydispersity in the bilayer thickness can be applied from the GUI.

Definition

The scattering intensity $I(q)$ for dilute, randomly oriented, "infinitely large" sheets or lamellae is

$$ I(q) = \text{scale}\frac{2\pi P(q)}{q^2\delta} + \text{background}
$$

The form factor is

$$ P(q) = \frac{2\Delta\rho^2}{q^2}(1-\cos(q\delta)) = \frac{4\Delta\rho^2}{q^2}\sin^2\left(\frac{q\delta}{2}\right)
$$
where $\delta$ is the total layer thickness and $\Delta\rho$ is the scattering length density difference.

This is the limiting form for a spherical shell of infinitely large radius. Note that the division by $\delta$ means that $scale$ in sasview is the volume fraction of sheet, $\phi = S\delta$ where $S$ is the area of sheet per unit volume. $S$ is half the Porod surface area per unit volume of a thicker layer (as that would include both faces of the sheet).

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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