# Lamellar Hg Stack Caille - lamellar_hg_stack_caille.py

```    ```# Note: model title and parameter table are inserted automatically
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.

The scattering intensity \$I(q)\$ is

.. math::

I(q) = 2 pi frac{P(q)S(q)}{q^2delta }

The form factor \$P(q)\$ is

.. math::

P(q) = frac{4}{q^2}ig{
Delta
ho_H left[sin[q(delta_H + delta_T)] - sin(qdelta_T)
ight]
+ Delta
ho_Tsin(qdelta_T)ig}^2

and the structure factor \$S(q)\$ is

.. math::

S(q) = 1 + 2 sum_1^{N-1}left(1-frac{n}{N}
ight)
cos(qdn)expleft(-frac{2q^2d^2alpha(n)}{2}
ight)

where

.. math::
:nowrap:

egin{align*}
alpha(n) &= frac{eta_{cp}}{4pi^2} left(ln(pi n)+gamma_E
ight)
&&  \
gamma_E  &= 0.5772156649
&& 	ext{Euler's constant} \
eta_{cp} &= frac{q_o^2k_B T}{8pisqrt{Koverline{B}}}
&& 	ext{Caille constant}
end{align*}

\$delta_T\$ is the tail length (or *length_tail*), \$delta_H\$ is the head
and \$Delta
ho_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.

Be aware that the computations may be very slow.

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 Reviewed by:**
"""

import numpy as np
from numpy import inf

name = "lamellar_hg_stack_caille"
description = """
[Random lamellar phase with Caille  structure factor]
randomly oriented stacks of infinite sheets
with Caille S(Q), having polydisperse spacing.
sld = Tail scattering length density
sld_solvent = solvent scattering length density
background = incoherent background
scale = scale factor
"""
category = "shape:lamellae"

single = False  # TODO: check
parameters = [
#   [ "name", "units", default, [lower, upper], "type",
#     "description" ],
["length_tail", "Ang", 10, [0, inf], "volume",
"Tail thickness"],
["length_head", "Ang", 2, [0, inf], "volume",
["Nlayers", "", 30, [1, inf], "",
"Number of layers"],
["d_spacing", "Ang", 40., [0.0, inf], "volume",
"lamellar d-spacing of Caille S(Q)"],
["Caille_parameter", "", 0.001, [0.0, 0.8], "",
"Caille parameter"],
["sld", "1e-6/Ang^2", 0.4, [-inf, inf], "sld",
"Tail scattering length density"],
["sld_head", "1e-6/Ang^2", 2.0, [-inf, inf], "sld",
["sld_solvent", "1e-6/Ang^2", 6, [-inf, inf], "sld",
"Solvent scattering length density"],
]

source = ["lamellar_hg_stack_caille.c"]

# No volume normalization despite having a volume parameter
# This should perhaps be volume normalized?
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)
length_head = thickness * np.random.uniform(0, 1)
Caille_parameter = np.random.uniform(0, 0.8)
pars = dict(
length_tail=length_tail,
Nlayers=Nlayers,
d_spacing=d_spacing,
Caille_parameter=Caille_parameter,
)
return pars

demo = dict(
scale=1, background=0,
Nlayers=20, d_spacing=200., Caille_parameter=0.05,