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r"""
Definition
----------
The scattering intensity $I(q)$ is calculated as
.. math::
I(q) = \begin{cases}
A q^{-m1} + \text{background} & q <= q_c \\
C q^{-m2} + \text{background} & q > q_c
\end{cases}
where $q_c$ = the location of the crossover from one slope to the other,
$A$ = the scaling coefficient that sets the overall intensity of the lower Q
power law region, $m1$ = power law exponent at low Q, and $m2$ = power law
exponent at high Q. The scaling of the second power law region (coefficient C)
is then automatically scaled to match the first by following formula:
.. math::
C = \frac{A q_c^{m2}}{q_c^{m1}}
.. note::
Be sure to enter the power law exponents as positive values!
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:** Wojciech Wpotrzebowski **Date:** February 18, 2016
* **Last Reviewed by:** Paul Butler **Date:** March 21, 2016
"""
import numpy as np
from numpy import inf, power, empty, errstate
name = "two_power_law"
title = "This model calculates an empirical functional form for SAS data \
characterized by two power laws."
description = """
I(q) = coef_A*pow(qval,-1.0*power1) + background for q<=q_c
=C*pow(qval,-1.0*power2) + background for q>q_c
where C=coef_A*pow(q_c,-1.0*power1)/pow(q_c,-1.0*power2).
coef_A = scaling coefficient
q_c = crossover location [1/A]
power_1 (=m1) = power law exponent at low Q
power_2 (=m2) = power law exponent at high Q
background = Incoherent background [1/cm]
"""
category = "shape-independent"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [
["coefficent_1", "", 1.0, [-inf, inf], "", "coefficent A in low Q region"],
["crossover", "1/Ang", 0.04,[0, inf], "", "crossover location"],
["power_1", "", 1.0, [0, inf], "", "power law exponent at low Q"],
["power_2", "", 4.0, [0, inf], "", "power law exponent at high Q"],
]
# pylint: enable=bad-whitespace, line-too-long
def Iq(q,
coefficent_1=1.0,
crossover=0.04,
power_1=1.0,
power_2=4.0,
):
"""
:param q: Input q-value (float or [float, float])
:param coefficent_1: Scaling coefficent at low Q
:param crossover: Crossover location
:param power_1: Exponent of power law function at low Q
:param power_2: Exponent of power law function at high Q
:return: Calculated intensity
"""
result = empty(q.shape, 'd')
index = (q <= crossover)
with errstate(divide='ignore'):
coefficent_2 = coefficent_1 * power(crossover, power_2 - power_1)
result[index] = coefficent_1 * power(q[index], -power_1)
result[~index] = coefficent_2 * power(q[~index], -power_2)
return result
Iq.vectorized = True # Iq accepts an array of q values
def random():
"""Return a random parameter set for the model."""
coefficient_1 = 1
crossover = 10**np.random.uniform(-3, -1)
power_1 = np.random.uniform(1, 6)
power_2 = np.random.uniform(1, 6)
pars = dict(
scale=1, #background=0,
coefficient_1=coefficient_1,
crossover=crossover,
power_1=power_1,
power_2=power_2,
)
return pars
tests = [
# Accuracy tests based on content in test/utest_extra_models.py
[{'coefficent_1': 1.0,
'crossover': 0.04,
'power_1': 1.0,
'power_2': 4.0,
'background': 0.0,
}, 0.001, 1000],
[{'coefficent_1': 1.0,
'crossover': 0.04,
'power_1': 1.0,
'power_2': 4.0,
'background': 0.0,
}, 0.150141, 0.125945],
[{'coefficent_1': 1.0,
'crossover': 0.04,
'power_1': 1.0,
'power_2': 4.0,
'background': 0.0,
}, 0.442528, 0.00166884],
[{'coefficent_1': 1.0,
'crossover': 0.04,
'power_1': 1.0,
'power_2': 4.0,
'background': 0.0,
}, (0.442528, 0.00166884), 0.00166884],
]
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