source: sasmodels/sasmodels/models/lamellarCaille.py @ ec2ca99

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)}{\delta q^2}

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) 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}

.. figure:: img/lamellarCaille_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 = "lamellarPS"
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
# 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,   [0, inf],   "",       "Number of layers"],
    ["spacing",          "Ang",      400.,  [0.0,inf],  "volume", "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], "",       "layer scattering length density"],
    ["solvent_sld",      "1e-6/Ang^2", 1.0, [-inf,inf], "",       "Solvent scattering length density"],
    ]
# pylint: enable=bad-whitespace, line-too-long

source = ["lamellarCaille_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,
            thickness=67., Nlayers=3.75, spacing=200.,
            Caille_parameter=0.268, sld=1.0, solvent_sld=6.34,
            thickness_pd=0.1, thickness_pd_n=100,
            spacing_pd=0.05, spacing_pd_n=40)

oldname = 'LamellarPSModel'
oldpars = dict(thickness='delta', Nlayers='N_plates',
               Caille_parameter='caille',
               sld='sld_bi', solvent_sld='sld_sol')
#
tests = [
    [{'scale': 1.0, 'background': 0.0, 'thickness': 30., 'Nlayers': 20.0,
      'spacing': 400., 'Caille_parameter': 0.1, 'sld': 6.3,
      'solvent_sld': 1.0, 'thickness_pd': 0.0, 'spacing_pd': 0.0},
     [0.001], [28895.13397]]
    ]