` ````
r"""
This model provides the form factor, $P(q)$, for a flexible cylinder
where the form factor is normalized by the volume of the cylinder.
**Inter-cylinder interactions are NOT provided for.**
.. math::
P(q) = ext{scale} left<F^2
ight>/V + ext{background}
where the averaging $left<ldots
ight>$ is applied only for the 1D
calculation
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}
Definitions
-----------
.. figure:: img/flexible_cylinder_geometry.jpg
The chain of contour length, $L$, (the total length) can be described as a
chain of some number of locally stiff segments of length $l_p$, the persistence
length (the length along the cylinder over which the flexible cylinder can be
considered a rigid rod).
The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain.
In the parameters, the sld and sld\_solvent represent the SLD of the cylinder
and solvent respectively.
Our model uses the form factor calculations in reference [1] as implemented in a
c-library provided by the NIST Center for Neutron Research (Kline, 2006). This states:
'Method 3 With Excluded Volume' is used.
The model is a parametrization of simulations of a discrete representation
of the worm-like chain model of Kratky and Porod applied in the
pseudocontinuous limit.
See equations (13,26-27) in the original reference for the details.
.. note::
There are several typos in the original reference that have been corrected
by WRC [2]. Details of the corrections are in the reference below. Most notably
- Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$
- Equations (23) and (24) are incorrect; WRC has entered these into
Mathematica and solved analytically. The results were then converted to
code.
- Equation (27) should be $q0 = max(a3/(Rg^2)^{1/2},3)$ instead of
$max(a3*b(Rg^2)^{1/2},3)$
- The scattering function is negative for a range of parameter values and
q-values that are experimentally accessible. A correction function has been
added to give the proper behavior.
**This is a model with complex behaviour depending on the ratio of** $L/b$ **and the
reader is strongly encouraged to read reference [1] before use.**
.. note::
There are several typos in the original reference that have been corrected
by WRC [2]. Details of the corrections are in the reference below. Most notably
- Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$
- Equations (23) and (24) are incorrect; WRC has entered these into
Mathematica and solved analytically. The results were then converted to
code.
- Equation (27) should be $q0 = max(a3/(Rg^2)^{1/2},3)$ instead of
$max(a3*b(Rg^2)^{1/2},3)$
- The scattering function is negative for a range of parameter values and
q-values that are experimentally accessible. A correction function has been
added to give the proper behavior.
**This is a model with complex behaviour depending on the ratio of** $L/b$ **and the
reader is strongly encouraged to read reference [1] before use.**
References
----------
.. [#] J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume effects.* Macromolecules, 29 (1996) 7602-7612
Correction of the formula can be found in
.. [#] W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from Cationic Wormlike Micelles.* Langmuir, 22(15) 2006 6539-6548
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:**
* **Last Reviewed by:** Steve King **Date:** March 26, 2019
"""
import numpy as np
from numpy import inf
name = "flexible_cylinder"
title = "Flexible cylinder where the form factor is normalized by the volume "
"of the cylinder."
description = """Note : scale and contrast = (sld - sld_solvent) are both
multiplicative factors in the model and are perfectly
correlated. One or both of these parameters must be held fixed
during model fitting.
"""
category = "shape:cylinder"
single = False # double precision only!
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [
["length", "Ang", 1000.0, [0, inf], "volume", "Length of the flexible cylinder"],
["kuhn_length", "Ang", 100.0, [0, inf], "volume", "Kuhn length of the flexible cylinder"],
["radius", "Ang", 20.0, [0, inf], "volume", "Radius of the flexible cylinder"],
["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Cylinder scattering length density"],
["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"],
]
# pylint: enable=bad-whitespace, line-too-long
source = ["lib/polevl.c", "lib/sas_J1.c", "lib/wrc_cyl.c", "flexible_cylinder.c"]
def random():
"""Return a random parameter set for the model."""
length = 10**np.random.uniform(2, 6)
radius = 10**np.random.uniform(1, 3)
kuhn_length = 10**np.random.uniform(-2, 0)*length
pars = dict(
length=length,
radius=radius,
kuhn_length=kuhn_length,
)
return pars
tests = [
# Accuracy tests based on content in test/utest_other_models.py
[{'length': 1000.0, # test T1
'kuhn_length': 100.0,
'radius': 20.0,
'sld': 1.0,
'sld_solvent': 6.3,
'background': 0.0001,
}, 0.001, 3509.2187],
# Additional tests with larger range of parameters
[{'length': 1000.0, # test T2
'kuhn_length': 100.0,
'radius': 20.0,
'sld': 1.0,
'sld_solvent': 6.3,
'background': 0.0001,
}, 1.0, 0.000595345],
[{'length': 10.0, # test T3
'kuhn_length': 800.0,
'radius': 2.0,
'sld': 6.0,
'sld_solvent': 12.3,
'background': 0.001,
}, 0.1, 1.55228],
[{'length': 100.0, # test T4
'kuhn_length': 800.0,
'radius': 50.0,
'sld': 0.1,
'sld_solvent': 5.1,
'background': 0.0,
}, 1.0, 0.000938456]
]
# There are a few branches in the code that ought to have test values:
#
# For length > 4 * kuhn_length
# if length > 10 * kuhn_length then C is scaled by 3.06 (L/b)^(-0.44)
# q*kuhn_length <= 3.1 => Sexv_new
# dS/dQ < 0 has different behaviour from dS/dQ >= 0
# T2 q*kuhn_length > 3.1 => a_long
#
# For length <= 4 * kuhn_length
# q*kuhn_length <= max(1.9/Rg_short, 3.0) => Sdebye((q*Rg)^2)
# q*Rg < 0.5 uses Pade approx, q*Rg > 1.0 uses math lib
# T3,T4 q*kuhn_length > max(1.9/Rg_short, 3.0) => a_short
#
# Note that the transitions between branches may be abrupt. You can see a
# several percent change around length=10*kuhn_length and length=4*kuhn_length
# using the following:
#
# sascomp flexible_cylinder -calc=double -sets=10 length=10*kuhn_length,10.000001*kuhn_length
# sascomp flexible_cylinder -calc=double -sets=10 length=4*kuhn_length,4.000001*kuhn_length
#
# The transition between low q and high q around q*kuhn_length = 3 seems
# to be good to 4 digits or better. This was tested by computing the value
# on each branches near the transition point and reporting the relative error
# for kuhn lengths of 10, 100 and 1000 and a variety of length:kuhn_length
# ratios.
```

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