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r"""
Definition
----------
This model calculates an empirical functional form for SAS data characterized
by a broad scattering peak. Many SAS spectra are characterized by a broad peak
even though they are from amorphous soft materials. For example, soft systems
that show a SAS peak include copolymers, polyelectrolytes, multiphase systems,
layered structures, etc.
The d-spacing corresponding to the broad peak is a characteristic distance
between the scattering inhomogeneities (such as in lamellar, cylindrical, or
spherical morphologies, or for bicontinuous structures).
The scattering intensity $I(q)$ is calculated as
.. math:: I(q) = \frac{A}{q^n} + \frac{C}{1 + (|q - q_0|\xi)^m}^p + B
Here the peak position is related to the d-spacing as $q_0 = 2\pi / d_0$.
$A$ is the Porod law scale factor, $n$ the Porod exponent, $C$ is the
Lorentzian scale factor, $m$ the exponent of $q$, $\xi$ the screening length,
and $B$ the flat background. $p$ generalizes the model. With m = 2 and p = 1
the Lorentz model is obtained whereas for m = 2 and p = 2 the Broad-Peak model
is identical to the Debye-Anderson-Brumberger (dab) 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
----------
None.
Authorship and Verification
----------------------------
* **Author:** NIST IGOR/DANSE **Date:** pre 2010
* **Last Modified by:** Dirk Honecker **Date:** May 28, 2021
* **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016
"""
import numpy as np
from numpy import inf, errstate
name = "broad_peak"
title = "Broad peak on top of a power law decay"
description = """\
I(q) = scale_p/pow(q,exponent)+scale_l/
pow(1.0 + pow(fabs(q-q_peak)*length_l,exponent_l), exponent_p )+ background
List of default parameters:
porod_scale = Porod term scaling
porod_exp = Porod exponent
peak_scale = Lorentzian term scaling
correlation_length = Correlation length [A]
peak_pos = peak location [1/A]
width_exp = peak width exponent
shape_exp = peak shape exponent
background = Incoherent background"""
category = "shape-independent"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [["porod_scale", "", 1.0e-05, [-inf, inf], "", "Power law scale factor"],
["porod_exp", "", 3.0, [-inf, inf], "", "Exponent of power law"],
["peak_scale", "", 10.0, [-inf, inf], "", "Scale factor for broad peak"],
["correlation_length", "Ang", 50.0, [-inf, inf], "", "screening length"],
["peak_pos", "1/Ang", 0.1, [-inf, inf], "", "Peak position in q"],
["width_exp", "", 2.0, [-inf, inf], "", "Exponent of peak width"],
["shape_exp", "", 1.0, [-inf, inf], "", "Exponent of peak shape"],
]
# pylint: enable=bad-whitespace, line-too-long
def Iq(q,
porod_scale=1.0e-5,
porod_exp=3.0,
peak_scale=10.0,
correlation_length=50.0,
peak_pos=0.1,
width_exp=2.0,
shape_exp=1.0):
"""
:param q: Input q-value
:param porod_scale: Power law scale factor
:param porod_exp: Exponent of power law
:param peak_scale: Scale factor for broad peak
:param correlation_length: Correlation length
:param peak_pos: Peak position in q
:param width_exp: Exponent of peak width
:param shape_exp: Exponent of peak shape
:return: Calculated intensity
"""
z = abs(q - peak_pos) * correlation_length
with errstate(divide='ignore'):
inten = (porod_scale / q ** porod_exp
+ peak_scale / (1 + z ** width_exp)) ** shape_exp
return inten
Iq.vectorized = True # Iq accepts an array of q values
def random():
"""Return a random parameter set for the model."""
pars = dict(
scale=1,
porod_scale=10**np.random.uniform(-8, -5),
porod_exp=np.random.uniform(1, 6),
peak_scale=10**np.random.uniform(0.3, 6),
correlation_length=10**np.random.uniform(0, 2),
peak_pos=10**np.random.uniform(-3, -1),
width_exp=np.random.uniform(1, 4),
shape_exp=np.random.uniform(1, 2),
)
pars['correlation_length'] /= pars['peak_pos']
pars['peak_scale'] *= pars['porod_scale'] / pars['peak_pos']**pars['porod_exp']
#pars['porod_scale'] = 0.
return pars
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