source: sasmodels/sasmodels/models/lamellarCailleHG.py @ eb69cce

# 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::

    %\begin{align*} % isn't working with pdflatex
    \begin{array}{rll}
    \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{array}


$\delta_T$ 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"

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')