Broad Peak - broad_peak.py

    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} + B

Here the peak position is related to the d-spacing as $q_0 = 2pi / d_0$.

$A$ 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.

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:** Paul kienle **Date:** July 24, 2016
* **Last Reviewed by:** Richard Heenan **Date:** March 21, 2016
"""

from numpy import inf, errstate

name = "broad_peak"
title = "Broad Lorentzian type peak on top of a power law decay"
description = """
      I(q) = scale_p/pow(q,exponent)+scale_l/
      (1.0 + pow((fabs(q-q_peak)*length_l),exponent_l) )+ background

      List of default parameters:
      porod_scale = Porod term scaling
      porod_exp = Porod exponent
      lorentz_scale = Lorentzian term scaling
      lorentz_length = Lorentzian screening length [A]
      peak_pos = peak location [1/A]
      lorentz_exp = Lorentzian 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"],
              ["lorentz_scale",  "",     10.0, [-inf, inf], "", "Scale factor for broad Lorentzian peak"],
              ["lorentz_length", "Ang",  50.0, [-inf, inf], "", "Lorentzian screening length"],
              ["peak_pos",       "1/Ang", 0.1, [-inf, inf], "", "Peak position in q"],
              ["lorentz_exp",    "",      2.0, [-inf, inf], "", "Exponent of Lorentz function"],
             ]
# pylint: enable=bad-whitespace, line-too-long

def Iq(q,
       porod_scale=1.0e-5,
       porod_exp=3.0,
       lorentz_scale=10.0,
       lorentz_length=50.0,
       peak_pos=0.1,
       lorentz_exp=2.0):
    """
    :param q:              Input q-value
    :param porod_scale:    Power law scale factor
    :param porod_exp:      Exponent of power law
    :param lorentz_scale:  Scale factor for broad Lorentzian peak
    :param lorentz_length: Lorentzian screening length
    :param peak_pos:       Peak position in q
    :param lorentz_exp:    Exponent of Lorentz function
    :return:               Calculated intensity
    """
    z = abs(q - peak_pos) * lorentz_length
    with errstate(divide='ignore'):
        inten = (porod_scale / q ** porod_exp
                 + lorentz_scale / (1 + z ** lorentz_exp))
    return inten
Iq.vectorized = True  # Iq accepts an array of q values

def random():
    import numpy as np
    pars = dict(
        scale=1,
        porod_scale=10**np.random.uniform(-8, -5),
        porod_exp=np.random.uniform(1, 6),
        lorentz_scale=10**np.random.uniform(0.3, 6),
        lorentz_length=10**np.random.uniform(0, 2),
        peak_pos=10**np.random.uniform(-3, -1),
        lorentz_exp=np.random.uniform(1, 4),
    )
    pars['lorentz_length'] /= pars['peak_pos']
    pars['lorentz_scale'] *= pars['porod_scale'] / pars['peak_pos']**pars['porod_exp']
    #pars['porod_scale'] = 0.
    return pars

demo = dict(scale=1, background=0,
            porod_scale=1.0e-05, porod_exp=3,
            lorentz_scale=10, lorentz_length=50, peak_pos=0.1, lorentz_exp=2)
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