Lamellar Hg - lamellar_hg.py
# Note: model title and parameter table are inserted automatically
r"""
This model provides the scattering intensity, $I(q)$, for a lyotropic lamellar
phase where a random distribution in solution are assumed. The SLD of the head
region is taken to be different from the SLD of the tail region.
Definition
----------
The scattering intensity $I(q)$ is
.. math::
I(q) = 2\pi\frac{\text{scale}}{2(\delta_H + \delta_T)} P(q) \frac{1}{q^2}
The form factor $P(q)$ is
.. math::
P(q) = \frac{4}{q^2}
\left\lbrace
\Delta \rho_H
\left[\sin[q(\delta_H + \delta_T)\ - \sin(q\delta_T)\right]
+ \Delta\rho_T\sin(q\delta_T)
\right\rbrace^2
where $\delta_T$ is *length_tail*, $\delta_H$ is *length_head*,
$\Delta\rho_H$ is the head contrast (*sld_head* $-$ *sld_solvent*),
and $\Delta\rho_T$ is tail contrast (*sld* $-$ *sld_solvent*).
The total thickness of the lamellar sheet is $\delta_H + \delta_T + \delta_T + \delta_H$.
Note that in a non aqueous solvent the chemical "head" group may be the
"Tail region" and vice-versa.
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 Modified by:**
* **Last Reviewed by:** S King and P Butler **Date** April 17, 2014
"""
import numpy as np
from numpy import inf
name = "lamellar_hg"
title = "Random lamellar phase with Head and Tail Groups"
description = """\
[Random lamellar phase with Head and Tail Groups]
I(q)= 2*pi*P(q)/(2(H+T)*q^(2)), where
P(q)= see manual
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"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [["length_tail", "Ang", 15, [0, inf], "volume", "Tail thickness ( total = H+T+T+H)"],
["length_head", "Ang", 10, [0, inf], "volume", "Head thickness"],
["sld", "1e-6/Ang^2", 0.4, [-inf,inf], "sld", "Tail scattering length density"],
["sld_head", "1e-6/Ang^2", 3.0, [-inf,inf], "sld", "Head scattering length density"],
["sld_solvent", "1e-6/Ang^2", 6, [-inf,inf], "sld", "Solvent scattering length density"]]
# pylint: enable=bad-whitespace, line-too-long
# No volume normalization despite having a volume parameter
# This should perhaps be volume normalized?
form_volume = """
return 1.0;
"""
Iq = """
const double qsq = q*q;
const double drh = sld_head - sld_solvent;
const double drt = sld - sld_solvent; //correction 13FEB06 by L.Porcar
const double qT = q*length_tail;
double Pq, inten;
Pq = drh*(sin(q*(length_head+length_tail))-sin(qT)) + drt*sin(qT);
Pq *= Pq;
Pq *= 4.0/(qsq);
inten = 2.0e-4*M_PI*Pq/qsq;
// normalize by the bilayer thickness
inten /= 2.0*(length_head+length_tail);
return inten;
"""
def random():
"""Return a random parameter set for the model."""
thickness = 10**np.random.uniform(1, 4)
length_head = thickness * np.random.uniform(0, 1)
length_tail = thickness - length_head
pars = dict(
length_head=length_head,
length_tail=length_tail,
)
return pars
demo = dict(scale=1, background=0,
length_tail=15, length_head=10,
sld=0.4, sld_head=3.0, sld_solvent=6.0,
length_tail_pd=0.2, length_tail_pd_n=40,
length_head_pd=0.01, length_head_pd_n=40)
#
tests = [
[{'scale': 1.0, 'background': 0.0, 'length_tail': 15.0, 'length_head': 10.0,
'sld': 0.4, 'sld_head': 3.0, 'sld_solvent': 6.0},
[0.001], [653143.9209]],
]
# ADDED by: RKH ON: 18Mar2016 converted from sasview previously, now renaming everything & sorting the docs
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