source: sasmodels/sasmodels/models/broad_peak.py @ 2c74c11

r"""
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).

Definition
----------

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 $q_o = 2\pi / d_o$.

$A$ 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.

*2013/09/09 - Description reviewed by King, S and Parker, P.*

"""

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

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)