Two Lorentzian - two_lorentzian.py

    r"""
Definition
----------

The scattering intensity $I(q)$ is calculated as

.. math::

    I(q) = frac{A}{1 +(Qxi_1)^n} + frac{C}{1 +(Qxi_2)^m} + 	ext{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


demo = dict(scale=1, background=0.1,
            lorentz_scale_1=10,
            lorentz_length_1=100.0,
            lorentz_exp_1=3.0,
            lorentz_scale_2=1,
            lorentz_length_2=10,
            lorentz_exp_2=2.0)

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],
    ]
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