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r"""
This model calculates the scattering from a gel structure,
but typically a physical rather than chemical network.
It is modeled as a sum of a low-q exponential decay (which happens to
give a functional form similar to Guinier scattering, so interpret with
care) plus a Lorentzian at higher-q values. See also the gel_fit model.
Definition
----------
The scattering intensity $I(q)$ is calculated as (Eqn. 5 from the reference)
.. math:: I(q) = I_G(0) \exp(-q^2\Xi ^2/2) + I_L(0)/(1+q^2\xi^2)
$\Xi$ is the length scale of the static correlations in the gel, which can
be attributed to the "frozen-in" crosslinks. $\xi$ is the dynamic correlation
length, which can be attributed to the fluctuating polymer chains between
crosslinks. $I_G(0)$ and $I_L(0)$ are the scaling factors for each of these
structures. Think carefully about how these map to your particular system!
.. note::
The peaked structure at higher $q$ values (Figure 2 from the reference)
is not reproduced by the model. Peaks can be introduced into the model
by summing this model with the :ref:`gaussian-peak` model.
For 2D data the 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
----------
#. G Evmenenko, E Theunissen, K Mortensen, H Reynaers,
*Polymer*, 42 (2001) 2907-2913
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**
"""
import numpy as np
from numpy import inf, exp
name = "gauss_lorentz_gel"
title = "Gauss Lorentz Gel model of scattering from a gel structure"
description = """
Class that evaluates a GaussLorentzGel model.
I(q) = scale_g*exp(- q^2*Z^2 / 2)+scale_l/(1+q^2*z^2)
+ background
List of default parameters:
scale_g = Gauss scale factor
Z = Static correlation length
scale_l = Lorentzian scale factor
z = Dynamic correlation length
background = Incoherent background
"""
category = "shape-independent"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [["gauss_scale", "", 100.0, [-inf, inf], "", "Gauss scale factor"],
["cor_length_static", "Ang", 100.0, [0, inf], "", "Static correlation length"],
["lorentz_scale", "", 50.0, [-inf, inf], "", "Lorentzian scale factor"],
["cor_length_dynamic", "Ang", 20.0, [0, inf], "", "Dynamic correlation length"],
]
# pylint: enable=bad-whitespace, line-too-long
def Iq(q,
gauss_scale=100.0,
cor_length_static=100.0,
lorentz_scale=50.0,
cor_length_dynamic=20.0):
"""
:param q: Input q-value
:param gauss_scale: Gauss scale factor
:param cor_length_static: Static correlation length
:param lorentz_scale: Lorentzian scale factor
:param cor_length_dynamic: Dynamic correlation length
:return: 1-D intensity
"""
term1 = gauss_scale *\
exp(-1.0*q*q*cor_length_static*cor_length_static/2.0)
term2 = lorentz_scale /\
(1.0+(q*cor_length_dynamic)*(q*cor_length_dynamic))
return term1 + term2
Iq.vectorized = True # Iq accepts an array of q values
def random():
"""Return a random parameter set for the model."""
gauss_scale = 10**np.random.uniform(1, 3)
lorentz_scale = 10**np.random.uniform(1, 3)
cor_length_static = 10**np.random.uniform(0, 3)
cor_length_dynamic = 10**np.random.uniform(0, 3)
pars = dict(
#background=0,
scale=1,
gauss_scale=gauss_scale,
lorentz_scale=lorentz_scale,
cor_length_static=cor_length_static,
cor_length_dynamic=cor_length_dynamic,
)
return pars
tests = [
# Accuracy tests based on content in test/utest_extra_models.py
[{'gauss_scale': 100.0,
'cor_length_static': 100.0,
'lorentz_scale': 50.0,
'cor_length_dynamic': 20.0,
}, 0.001, 149.482],
[{'gauss_scale': 100.0,
'cor_length_static': 100.0,
'lorentz_scale': 50.0,
'cor_length_dynamic': 20.0,
}, 0.105363, 9.1913],
[{'gauss_scale': 100.0,
'cor_length_static': 100.0,
'lorentz_scale': 50.0,
'cor_length_dynamic': 20.0,
}, 0.441623, 0.633811],
# Additional tests with larger range of parameters
[{'gauss_scale': 10.0,
'cor_length_static': 100.0,
'lorentz_scale': 3.0,
'cor_length_dynamic': 1.0,
}, 0.1, 2.9712970297],
[{'gauss_scale': 10.0,
'cor_length_static': 100.0,
'lorentz_scale': 3.0,
'cor_length_dynamic': 1.0,
'background': 100.0
}, 5.0, 100.116384615],
[{'gauss_scale': 10.0,
'cor_length_static': 100.0,
'lorentz_scale': 3.0,
'cor_length_dynamic': 1.0,
'background': 0.0,
}, 200., 7.49981250469e-05],
]
|
