source: sasmodels/sasmodels/models/unified_power_Rg.py @ cdcebf1

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

This model employs the empirical multiple level unified Exponential/Power-law 
fit method developed by Beaucage. Four functions are included so that 1, 2, 3, 
or 4 levels can be used. In addition a 0 level has been added which simply 
calculates

.. math::

    I(q) = \text{scale} / q + \text{background}

The Beaucage method is able to reasonably approximate the scattering from
many different types of particles, including fractal clusters, random coils
(Debye equation), ellipsoidal particles, etc.

The model works best for mass fractal systems characterized by Porod exponents 
between 5/3 and 3. It should not be used for surface fractal systems. Hammouda 
(2010) has pointed out a deficiency in the way this model handles the 
transitioning between the Guinier and Porod regimes and which can create 
artefacts that appear as kinks in the fitted model function.

Also see the Guinier_Porod model.

The empirical fit function is:

.. math::

    I(q) = \text{background}
    + \sum_{i=1}^N \Bigl[
        G_i \exp\Bigl(-\frac{q^2R_{gi}^2}{3}\Bigr)
       + B_i \exp\Bigl(-\frac{q^2R_{g(i+1)}^2}{3}\Bigr)
             \Bigl(\frac{1}{q_i^*}\Bigr)^{P_i} \Bigr]

where

.. math::

    q_i^* = \frac{q}{\operatorname{erf}^3(q R_{gi}/\sqrt{6}}


For each level, the four parameters $G_i$, $R_{gi}$, $B_i$ and $P_i$ must
be chosen.  Beaucage has an additional factor $k$ in the definition of
$q_i^*$ which is ignored here.

For example, to approximate the scattering from random coils (Debye equation),
set $R_{gi}$ as the Guinier radius, $P_i = 2$, and $B_i = 2 G_i / R_{gi}$

See the references for further information on choosing the parameters.

For 2D data: The 2D 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
----------

G Beaucage, *J. Appl. Cryst.*, 28 (1995) 717-728

G Beaucage, *J. Appl. Cryst.*, 29 (1996) 134-146

B Hammouda, *Analysis of the Beaucage model, J. Appl. Cryst.*, (2010), 43, 1474-1478

"""

from __future__ import division

import numpy as np
from numpy import inf, exp, sqrt, errstate
from scipy.special import erf

category = "shape-independent"
name = "unified_power_Rg"
title = "Unified Power Rg"
description = """
        The Beaucage model employs the empirical multiple level unified
        Exponential/Power-law fit method developed by G. Beaucage. Four functions
        are included so that 1, 2, 3, or 4 levels can be used.
        """

# pylint: disable=bad-whitespace, line-too-long
parameters = [
    ["level",     "",     1,      [0, 6], "", "Level number"],
    ["rg[level]", "Ang",  15.8,   [0, inf], "", "Radius of gyration"],
    ["power[level]", "",  4,      [-inf, inf], "", "Power"],
    ["B[level]",  "1/cm", 4.5e-6, [-inf, inf], "", ""],
    ["G[level]",  "1/cm", 400,    [0, inf], "", ""],
    ]
# pylint: enable=bad-whitespace, line-too-long

def Iq(q, level, rg, power, B, G):
    ilevel = int(level)
    if ilevel == 0:
        with errstate(divide='ignore'):
            return 1./q

    with errstate(divide='ignore', invalid='ignore'):
        result = np.zeros(q.shape, 'd')
        for i in range(ilevel):
            exp_now = exp(-(q*rg[i])**2/3.)
            pow_now = (erf(q*rg[i]/sqrt(6.))**3/q)**power[i]
            if i < ilevel-1:
                exp_next = exp(-(q*rg[i+1])**2/3.)
            else:
                exp_next = 1
            result += G[i]*exp_now + B[i]*exp_next*pow_now

    result[q == 0] = np.sum(G[:ilevel])
    return result

Iq.vectorized = True

demo = dict(
    level=2,
    rg=[15.8, 21],
    power=[4, 2],
    B=[4.5e-6, 0.0006],
    G=[400, 3],
    scale=1.,
    background=0.,
)