Lamellar Stack Caille - lamellar_stack_caille.py
r"""
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
.. math::
I(q) = 2\pi \frac{P(q)S(q)}{q^2\delta }
The form factor is
.. math::
P(q) = \frac{2\Delta\rho^2}{q^2}\left(1-\cos q\delta \right)
and the structure factor is
.. math::
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
.. math::
:nowrap:
\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
.. math::
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
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**
"""
import numpy as np
from numpy import inf
name = "lamellar_stack_caille"
title = "Random lamellar sheet with Caille structure factor"
description = """\
[Random lamellar phase with Caille structure factor]
randomly oriented stacks of infinite sheets
with Caille S(Q), having polydisperse spacing.
sld = sheet scattering length density
sld_solvent = solvent scattering length density
background = incoherent background
scale = scale factor
"""
category = "shape:lamellae"
single = False # TODO: check
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [
["thickness", "Ang", 30.0, [0, inf], "volume", "sheet thickness"],
["Nlayers", "", 20, [1, inf], "", "Number of layers"],
["d_spacing", "Ang", 400., [0.0,inf], "volume", "lamellar d-spacing of Caille S(Q)"],
["Caille_parameter", "1/Ang^2", 0.1, [0.0,0.8], "", "Caille parameter"],
["sld", "1e-6/Ang^2", 6.3, [-inf,inf], "sld", "layer scattering length density"],
["sld_solvent", "1e-6/Ang^2", 1.0, [-inf,inf], "sld", "Solvent scattering length density"],
]
# pylint: enable=bad-whitespace, line-too-long
source = ["lamellar_stack_caille.c"]
def random():
"""Return a random parameter set for the model."""
total_thickness = 10**np.random.uniform(2, 4.7)
Nlayers = np.random.randint(2, 200)
d_spacing = total_thickness / Nlayers
thickness = d_spacing * np.random.uniform(0, 1)
Caille_parameter = np.random.uniform(0, 0.8)
pars = dict(
thickness=thickness,
Nlayers=Nlayers,
d_spacing=d_spacing,
Caille_parameter=Caille_parameter,
)
return pars
# No volume normalization despite having a volume parameter
# This should perhaps be volume normalized?
form_volume = """
return 1.0;
"""
demo = dict(scale=1, background=0,
thickness=67., Nlayers=3.75, d_spacing=200.,
Caille_parameter=0.268, sld=1.0, sld_solvent=6.34,
thickness_pd=0.1, thickness_pd_n=100,
d_spacing_pd=0.05, d_spacing_pd_n=40)
#
tests = [
[{'scale': 1.0, 'background': 0.0, 'thickness': 30., 'Nlayers': 20.0,
'd_spacing': 400., 'Caille_parameter': 0.1, 'sld': 6.3,
'sld_solvent': 1.0, 'thickness_pd': 0.0, 'd_spacing_pd': 0.0},
[0.001], [28895.13397]]
]
# ADDED by: RKH ON: 18Mar2016 converted from sasview previously, now renaming everything & sorting the docs
Back to Model
Download