source: sasmodels/sasmodels/models/lamellarCailleHG.py @ 56fac50

# 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)}{\delta q^2}


The form factor $P(q)$ is

.. math::

        P(q) = \frac{4}{q^2}\big\{
        \Delta\rho_H \left[\sin[q(\delta_H + \delta_T)] - \sin(q\delta_T)\right]
            + \Delta\rho_T\sin(q\delta_T)\big\}^2

and the structure factor $S(q)$ 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*}


$\delta_T$ 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

.. math::

    q = \sqrt{q_x^2 + q_y^2}

.. figure:: img/lamellarCailleHG_1d.jpg

    1D plot using the default values (w/6000 data point).

References
----------

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
"""
from numpy import inf

name = "lamellarCailleHG"
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.
        layer thickness =(H+T+T+H) = 2(Head+Tail)
        sld = Tail scattering length density
        sld_head = Head scattering length density
        sld_solvent = solvent scattering length density
        background = incoherent background
        scale = scale factor
"""
category = "shape:lamellae"

single = False
parameters = [
    #   [ "name", "units", default, [lower, upper], "type",
    #     "description" ],
    ["tail_length", "Ang", 10, [0, inf], "volume",
     "Tail thickness"],
    ["head_length", "Ang", 2, [0, inf], "volume",
     "head thickness"],
    ["Nlayers", "", 30, [0, inf], "",
     "Number of layers"],
    ["spacing", "Ang", 40., [0.0, inf], "volume",
     "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], "",
     "Tail scattering length density"],
    ["head_sld", "1e-6/Ang^2", 2.0, [-inf, inf], "",
     "Head scattering length density"],
    ["solvent_sld", "1e-6/Ang^2", 6, [-inf, inf], "",
     "Solvent scattering length density"],
    ]

source = ["lamellarCailleHG_kernel.c"]

# No volume normalization despite having a volume parameter
# This should perhaps be volume normalized?
form_volume = """
    return 1.0;
    """

Iqxy = """
    return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS);
    """

# ER defaults to 0.0
# VR defaults to 1.0

demo = dict(
    scale=1, background=0,
    Nlayers=20, spacing=200., Caille_parameter=0.05,
    tail_length=15, head_length=10,
    #sld=-1, head_sld=4.0, solvent_sld=6.0,
    sld=-1, head_sld=4.1, solvent_sld=6.0,
    tail_length_pd=0.1, tail_length_pd_n=20,
    head_length_pd=0.05, head_length_pd_n=30,
    spacing_pd=0.2, spacing_pd_n=40,
    )

oldname = 'LamellarPSHGModel'

oldpars = dict(
    tail_length='deltaT', head_length='deltaH', Nlayers='n_plates',
    Caille_parameter='caille', sld='sld_tail', head_sld='sld_head',
    solvent_sld='sld_solvent')
#
tests = [[{'scale': 1.0, 'background': 0.0, 'tail_length': 10.0, 'head_length': 2.0,
           'Nlayers': 30.0, 'spacing': 40., 'Caille_parameter': 0.001, 'sld': 0.4,
           'head_sld': 2.0, 'solvent_sld': 6.0, 'tail_length_pd': 0.0,
           'head_length_pd': 0.0, 'spacing_pd': 0.0}, [0.001], [6838238.571488]]]