source: sasmodels/sasmodels/models/broad_peak.py @ f57d123

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).

The returned value is scaled to units of |cm^-1|, absolute scale.

Definition
----------

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

.. math:

    I(q) = \frac{A}{Q^n} + \frac{C}{1 + (Q\xi}^m} + B

Here the peak position is related to the d-spacing as *Q0* = 2|pi| / *d0*.

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}

=====================  =========  =============
Parameter name          Units     Default value
=====================  =========  =============
lorentz_scale(=C)       None      10
porod_scale  (=A)       None      1e-05
lorentz_length (= |xi| )  |Ang|     50
peak_pos  (=Q0)         |Ang^-1|  0.1
porod_exp  (=n)         None      2
lorentz_exp (=m)        None      3
Background (=B)        |cm^-1|   0.1
==================  ========  =============

.. image:: img/image175.jpg

*Figure. 1D plot using the default values (w/200 data point).*

REFERENCE

None.

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

"""

import numpy as np
from numpy import pi, inf, sin, cos, sqrt, exp, log, fabs

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"""

parameters = [
#   [ "name", "units", default, [lower, upper], "type",
#     "description" ],

    [ "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 postion in q" ],
    [ "lorentz_exp", "",  2.0, [-inf, inf], "",
      "exponent of Lorentz function" ],
    ]


def form_volume():
    return 1

def Iq(q, porod_scale, porod_exp, lorentz_scale, lorentz_length, peak_pos, lorentz_exp):
    inten = porod_scale/pow(q,porod_exp) + lorentz_scale/(1.0 \
        + pow((fabs(q-peak_pos)*lorentz_length),lorentz_exp))
    return inten  

# FOR VECTORIZED VERSION, UNCOMMENT THE NEXT LINE
Iq.vectorized = True

def Iqxy(qx, qy, porod_scale, porod_exp, lorentz_scale, lorentz_length, peak_pos, lorentz_exp):
    return Iq(sqrt(qx**2 + qy**2), porod_scale, porod_exp, lorentz_scale, lorentz_length, peak_pos, lorentz_exp)

# FOR VECTORIZED VERSION, UNCOMMENT THE NEXT LINE
Iqxy.vectorized = True


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,
    )
oldname = "BroadPeakModel"
oldpars = dict(porod_scale='scale_p', porod_exp='exponent_p', 
        lorentz_scale='scale_l', lorentz_length='length_l', peak_pos='q_peak', 
        lorentz_exp='exponent_l', scale=None)