lamellar

Lyotropic lamellar phase with uniform SLD and random distribution

Parameter Description Units Default value
scale Scale factor or Volume fraction None 1
background Source background cm-1 0.001
thickness total layer thickness Å 50
sld Layer scattering length density 10-6Å-2 1
sld_solvent Solvent scattering length density 10-6Å-2 6

The returned value is scaled to units of cm-1 sr-1, absolute scale.

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

\[q = \sqrt{q_x^2 + q_y^2}\]
../../_images/lamellar_autogenfig.png

Fig. 46 1D plot corresponding to the default parameters of the model.

Source

lamellar.py

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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