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r"""
Definition
----------
The scattering intensity $I(q)$ is calculated as
.. math::
I(q) = \frac{A}{1 +(Q\xi_1)^n} + \frac{C}{1 +(Q\xi_2)^m} + \text{B}
where $A$ = Lorentzian scale factor #1, $C$ = Lorentzian scale #2,
$\xi_1$ and $\xi_2$ are the corresponding correlation lengths, and $n$ and
$m$ are the respective power law exponents (set $n = m = 2$ for
Ornstein-Zernicke behaviour).
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
----------
None.
Authorship and Verification
----------------------------
* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Piotr rozyczko **Date:** January 29, 2016
* **Last Reviewed by:** Paul Butler **Date:** March 21, 2016
"""
import numpy as np
from numpy import inf, power
name = "two_lorentzian"
title = "This model calculates an empirical functional form for SAS data \
characterized by two Lorentzian-type functions."
description = """I(q) = scale_1/(1.0 + pow((q*length_1),exponent_1))
+ scale_2/(1.0 + pow((q*length_2),exponent_2) )+ background
scale_1 = Lorentzian term scaling #1
length_1 = Lorentzian screening length #1 [A]
exponent_1 = Lorentzian exponent #1
scale_2 = Lorentzian term scaling #2
length_2 = Lorentzian screening length #2 [A]
exponent_2 = Lorentzian exponent #2
background = Incoherent background
"""
category = "shape-independent"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [["lorentz_scale_1", "", 10.0, [-inf, inf], "", "First power law scale factor"],
["lorentz_length_1", "Ang", 100.0, [-inf, inf], "", "First Lorentzian screening length"],
["lorentz_exp_1", "", 3.0, [-inf, inf], "", "First exponent of power law"],
["lorentz_scale_2", "", 1.0, [-inf, inf], "", "Second scale factor for broad Lorentzian peak"],
["lorentz_length_2", "Ang", 10.0, [-inf, inf], "", "Second Lorentzian screening length"],
["lorentz_exp_2", "", 2.0, [-inf, inf], "", "Second exponent of power law"],
]
# pylint: enable=bad-whitespace, line-too-long
def Iq(q,
lorentz_scale_1=10.0,
lorentz_length_1=100.0,
lorentz_exp_1=3.0,
lorentz_scale_2=1.0,
lorentz_length_2=10.0,
lorentz_exp_2=2.0):
"""
:param q: Input q-value (float or [float, float])
:param lorentz_scale_1: Second scale factor for broad Lorentzian peak
:param lorentz_length_1: First Lorentzian screening length
:param lorentz_exp_1: Exponent of the second Lorentz function
:param lorentz_scale_2: Second scale factor for broad Lorentzian peak
:param lorentz_length_2: Second Lorentzian screening length
:param lorentz_exp_2: Exponent of the second Lorentz function
:return: Calculated intensity
"""
# pylint: disable=bad-whitespace
intensity = lorentz_scale_1/(1.0 +
power(q*lorentz_length_1, lorentz_exp_1))
intensity += lorentz_scale_2/(1.0 +
power(q*lorentz_length_2, lorentz_exp_2))
# pylint: enable=bad-whitespace
return intensity
Iq.vectorized = True # Iq accepts an array of q values
def random():
"""Return a random parameter set for the model."""
scale = 10**np.random.uniform(0, 4, 2)
length = 10**np.random.uniform(1, 4, 2)
expon = np.random.uniform(1, 6, 2)
pars = dict(
#background=0,
scale=1, # scale provided in model
lorentz_scale_1=scale[0],
lorentz_length_1=length[0],
lorentz_exp_1=expon[0],
lorentz_scale_2=scale[1],
lorentz_length_2=length[1],
lorentz_exp_2=expon[1],
)
return pars
tests = [
# Accuracy tests based on content in test/utest_extra_models.py
[{'lorentz_scale_1': 10.0,
'lorentz_length_1': 100.0,
'lorentz_exp_1': 3.0,
'lorentz_scale_2': 1.0,
'lorentz_length_2': 10.0,
'lorentz_exp_2': 2.0,
'background': 0.1,
}, 0.001, 11.08991],
[{'lorentz_scale_1': 10.0,
'lorentz_length_1': 100.0,
'lorentz_exp_1': 3.0,
'lorentz_scale_2': 1.0,
'lorentz_length_2': 10.0,
'lorentz_exp_2': 2.0,
'background': 0.1,
}, 0.150141, 0.410245],
[{'lorentz_scale_1': 10.0,
'lorentz_length_1': 100.0,
'lorentz_exp_1': 3.0,
'lorentz_scale_2': 1.0,
'lorentz_length_2': 10.0,
'lorentz_exp_2': 2.0,
'background': 0.1,
}, 0.442528, 0.148699],
# Additional tests with larger range of parameters
[{'lorentz_scale_1': 10.0,
'lorentz_length_1': 100.0,
'lorentz_exp_1': 3.0,
'lorentz_scale_2': 1.0,
'lorentz_length_2': 10.0,
'lorentz_exp_2': 2.0,
}, 0.000332070182643, 10.9996228107],
[{'lorentz_scale_1': 0.0,
'lorentz_length_1': 0.0,
'lorentz_exp_1': 0.0,
'lorentz_scale_2': 0.0,
'lorentz_length_2': 0.0,
'lorentz_exp_2': 0.0,
'background': 100.0
}, 5.0, 100.0],
[{'lorentz_scale_1': 200.0,
'lorentz_length_1': 10.0,
'lorentz_exp_1': 0.1,
'lorentz_scale_2': 0.1,
'lorentz_length_2': 5.0,
'lorentz_exp_2': 2.0
}, 20000., 45.5659201896],
]
|
