2.1.3.3. Lamellarcaillehg¶
Random lamellar sheet with Caille structure factor
Parameter | Description | Units | Default value |
---|---|---|---|
scale | Source intensity | None | 1 |
background | Source background | cm-1 | 0 |
tail_length | Tail thickness | Å | 10 |
head_length | head thickness | Å | 2 |
Nlayers | Number of layers | None | 30 |
spacing | 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 |
head_sld | Head scattering length density | 10-6Å-2 | 2 |
solvent_sld | Solvent scattering length density | 10-6Å-2 | 6 |
The returned value is scaled to units of cm-1.
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
The form factor \(P(q)\) is
and the structure factor \(S(q)\) is
where
\(\delta_T\) is the tail length (or tail_length), \(\delta_H\) is the head thickness (or head_length), \(\Delta\rho_H\) 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.
The 2D scattering intensity is calculated in the same way as 1D, where the \(q\) vector is defined as
The returned value is in units of cm-1, on absolute scale.
Figure. 1D plot using the default values (w/6000 data point).
Our model uses the form factor calculations implemented in a C library provided by the NIST Center for Neutron Research (Kline, 2006).
REFERENCE
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