` ````
# 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 outer,
solvent-exposed region is taken to be different from the SLD of the inner region.
This model is intended to be used with input parameters including the molecualr
volumes and bound coherrent scattering length of the inner portion of the
amphiphile (solvent not exposed), the outer portion of the amphiphile (outer
solvent exposed), and solvent molecule.
The variable parameters are intended to be the APL, the average area per
amphiphile (lipid) molecules at the inner/outer interface and n_W, the number
of solvent (water) molecules residing in the outer region of the bilayer.
This model is adapted from the lipid bilayer model of Nagle and Wiener(1988) and fitting function lamellar_hg.
Note that the model can be applied with other combinations of input parameters/
assumptions; and is ideally applied as a simultaneous fit to datasets with
multiple independent measurments; such as neutron contrast variation strategies.
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_tail* $-$ *sld_solvent*),
$\length_tail$ equals (*Volume_tail*$/$*APL*),
$\length_head$ equals ((*Volume_head*$+$*n_H*$*$*Volume_water*)$/$*APL*),
$\sld_head$ equals (*B_head*$+$*n_H*$*$*B_water*)$/$(*Volume_head*$+$*n_H*$*$*Volume_water*),
$\sld_tails equals (*B_tail*$/$*Volume_tail*).
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
.. [#] Nagle, J., & Wiener, M. (1988). Structure of fully hydrated bilayer dispersions. Biochimica et Biophysica Acta (BBA)-Biomembranes, 942(1), 1-10
.. [#] Tan, L., Elkins J.G., Davison, B.H., Kelly, E.G., Nickels, J.D. Implementation of Slab Model of Bilayer Structure in SASview. Submitted (2020)
Authorship and Verification
----------------------------
* **Author:luoxi Tan, James G. Elkins, Brian H. Davison, Elizabeth G. Kelly and Jonathan D. Nickels
"""
import numpy as np
from numpy import inf
name = "lamellar_Slab_APL_nW"
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_c = Tail scattering length density
SLD_h = Head scattering length density
SLD_w = solvent scattering length density
background = incoherent background
scale = scale factor
B_h = Bound coherrent scattering length of lipid headgroup
B_c = Bound coherrent scattering length of lipid tails
B_w = Bound coherrent scattering length of water
APL = Average area per lipid
V_w = Molecualar volume of water
V_h = Molecular volume of lipid headgroup
V_c = Molecualr volume of lipid tails
N_w = Number molecules of water molecule permeating into bilayer region
Dc = Average length of lipid tail
Dh = Average length of lipid headgroup
"""
category = "shape:lamellae"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type","description"],
parameters = [["B_h", "1e-6*Ang", 58, [-inf, inf], "volume", "Bound coherrent scattering length of lipid headgroup"],
["B_c", "1e-6*Ang", -9.05984, [-inf, inf], "volume", "Bound coherrent scattering length of lipid tails"],
["B_w", "1e-6*Ang", 19.145, [-inf,inf], "volume", "Bound coherrent scattering length of water"],
["APL", "Ang^2", 60, [0,inf], "volume", "Area"],
["V_w", "Ang^3", 30.4, [0,inf], "volume", "Molecualar volume of water"],
["V_h", "Ang^3", 211, [0,inf], "volume", "Molecular volume of lipid headgroup"],
["V_c", "Ang^3", 896.608, [0,inf], "volume", "Molecualr volume of lipid tails"],
["N_w", "None", 20, [0,inf], "sld", "Number molecules of water molecule permeating into bilayer region"]
]
# pylint: enable=bad-whitespace, line-too-long
# No volume normalization despite having a volume parameter
# This should perhaps be volume normalized?
def Iq(q,
B_h=58,
B_c=-9.05984,
B_w=19.145,
APL=60,
V_w=30.4,
V_h=211,
V_c=896.608,
N_w=20):
SLD_h = (B_h + N_w*B_w)/(V_h +N_w*V_w) #the calculation of sld of headgroup
SLD_c = (B_c/V_c) #the calculation of sld of tail
SLD_w = (B_w/V_w) #the calculation of sld of solvent
Dc = (V_c)/(APL) #the calculation of length of lipid hydrocarbon chains
Dh = (V_h + N_w*V_w)/APL #the calculation of length of lipid headgroup
qsq = q*q # q square
drh = (SLD_h - SLD_w) #delta rho_H (the head contrast)
drt = (SLD_c - SLD_w) #delat rho_T (the tail contrast)
qT = q*Dc
Pq = drh*(np.sin(q*(Dh + Dc))-np.sin(qT)) + drt*np.sin(qT)
Pq *= Pq
Pq *=4.0/(qsq)
inten = 2*np.pi*Pq/qsq
inten /=2.0*(Dc +Dh)
return inten
def random(): #the random function which can generate random number for the vriable parameters
"""Return a random parameter set for the model."""
APL = np.random.uniform(1, 500)
N_w = np.random.uniform(0, 100)
pars = dict(
APL = APL,
N_w = N_w
)
return pars
```

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