Lamellar Stack Paracrystal - lamellar_stack_paracrystal.py
# Note: model title and parameter table are inserted automatically
r"""
This model calculates the scattering from a stack of repeating lamellar
structures. The stacks of lamellae (infinite in lateral dimension) are
treated as a paracrystal to account for the repeating spacing. The repeat
distance is further characterized by a Gaussian polydispersity. **This model
can be used for large multilamellar vesicles.**
Definition
----------
In the equations below,
- *scale* is used instead of the mass per area of the bilayer $\Gamma_m$
(this corresponds to the volume fraction of the material in the bilayer,
*not* the total excluded volume of the paracrystal),
- *sld* $-$ *sld_solvent* is the contrast $\Delta \rho$,
- *thickness* is the layer thickness $t$,
- *Nlayers* is the number of layers $N$,
- *d_spacing* is the average distance between adjacent layers
$\langle D \rangle$, and
- *sigma_d* is the relative standard deviation of the Gaussian
layer distance distribution $\sigma_D / \langle D \rangle$.
The scattering intensity $I(q)$ is calculated as
.. math::
I(q) = 2\pi\Delta\rho^2\Gamma_m\frac{P_\text{bil}(q)}{q^2} Z_N(q)
The form factor of the bilayer is approximated as the cross section of an
infinite, planar bilayer of thickness $t$ (compare the equations for the
lamellar model).
.. math::
P_\text{bil}(q) = \left(\frac{\sin(qt/2)}{qt/2}\right)^2
$Z_N(q)$ describes the interference effects for aggregates
consisting of more than one bilayer. The equations used are (3-5)
from the Bergstrom reference:
.. math::
Z_N(q) = \frac{1 - w^2}{1 + w^2 - 2w \cos(q \langle D \rangle)}
+ x_N S_N + (1 - x_N) S_{N+1}
where
.. math::
S_N(q) = \frac{a_N}{N}[1 + w^2 - 2 w \cos(q \langle D \rangle)]^2
and
.. math::
a_N &= 4w^2 - 2(w^3 + w) \cos(q \langle D \rangle) \\
&\quad - 4w^{N+2}\cos(Nq \langle D \rangle)
+ 2 w^{N+3}\cos[(N-1)q \langle D \rangle]
+ 2w^{N+1}\cos[(N+1)q \langle D \rangle]
for the layer spacing distribution $w = \exp(-\sigma_D^2 q^2/2)$.
Non-integer numbers of stacks are calculated as a linear combination of
the lower and higher values
.. math::
N_L = x_N N + (1 - x_N)(N+1)
The 2D scattering intensity is the same as 1D, regardless of the orientation
of the $q$ vector which is defined as
.. math::
q = \sqrt{q_x^2 + q_y^2}
Reference
---------
.. [#] M Bergstrom, J S Pedersen, P Schurtenberger, S U Egelhaaf, *J. Phys. Chem. B*, 103 (1999) 9888-9897
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**
"""
import numpy as np
from numpy import inf
name = "lamellar_stack_paracrystal"
title = "Random lamellar sheet with paracrystal structure factor"
description = """\
[Random lamellar phase with paracrystal structure factor]
randomly oriented stacks of infinite sheets
with paracrytal 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
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [["thickness", "Ang", 33.0, [0, inf], "volume",
"sheet thickness"],
["Nlayers", "", 20, [1, inf], "",
"Number of layers"],
["d_spacing", "Ang", 250., [0.0, inf], "",
"lamellar spacing of paracrystal stack"],
["sigma_d", "Ang", 0.0, [0.0, inf], "",
"Sigma (polydispersity) of the lamellar spacing"],
["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld",
"layer scattering length density"],
["sld_solvent", "1e-6/Ang^2", 6.34, [-inf, inf], "sld",
"Solvent scattering length density"],
]
source = ["lamellar_stack_paracrystal.c"]
form_volume = """
return 1.0;
"""
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)
# Let polydispersity peak around 15%; 95% < 0.4; max=100%
sigma_d = np.random.beta(1.5, 7)
pars = dict(
thickness=thickness,
Nlayers=Nlayers,
d_spacing=d_spacing,
sigma_d=sigma_d,
)
return pars
demo = dict(scale=1, background=0,
thickness=33, Nlayers=20, d_spacing=250, sigma_d=0.2,
sld=1.0, sld_solvent=6.34,
thickness_pd=0.2, thickness_pd_n=40)
#
tests = [
[{'scale': 1.0, 'background': 0.0, 'thickness': 33., 'Nlayers': 20.0,
'd_spacing': 250., 'sigma_d': 0.2, 'sld': 1.0,
'sld_solvent': 6.34, 'thickness_pd': 0.0, 'thickness_pd_n': 40},
[0.001, 0.215268], [21829.3, 0.00487686]],
]
# ADDED by: RKH ON: 18Mar2016 converted from sasview previously,
# now renaming everything & sorting the docs
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