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

#
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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