Dab - dab.py
r"""
Calculates the scattering from a randomly distributed, two-phase system based on
the Debye-Anderson-Brumberger (DAB) model for such systems. The two-phase system
is characterized by a single length scale, the correlation length, which is a
measure of the average spacing between regions of phase 1 and phase 2. **The
model also assumes smooth interfaces between the phases** and hence exhibits
Porod behavior $(I \sim q^{-4})$ at large $q$, $(qL \gg 1)$.
The DAB model is ostensibly a development of the earlier Debye-Bueche model.
Definition
----------
.. math::
I(q) = \text{scale}\cdot\frac{L^3}{(1 + (q\cdot L)^2)^2} + \text{background}
where scale is
.. math:: \text{scale} = 8 \pi \phi (1-\phi) \Delta\rho^2
and the parameter $L$ is the correlation length.
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
----------
.. [#] P Debye, H R Anderson, H Brumberger, *Scattering by an Inhomogeneous Solid. II. The Correlation Function and its Application*, *J. Appl. Phys.*, 28(6) (1957) 679
.. [#] P Debye, A M Bueche, *Scattering by an Inhomogeneous Solid*, *J. Appl. Phys.*, 20 (1949) 518
Source
------
`dab.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/dab.py>`_
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:**
* **Last Reviewed by:** Steve King & Peter Parker **Date:** September 09, 2013
* **Source added by :** Steve King **Date:** March 25, 2019
"""
import numpy as np
from numpy import inf
name = "dab"
title = "DAB (Debye Anderson Brumberger) Model"
description = """\
F(q)= scale * L^3/(1 + (q*L)^2)^2
L: the correlation length
"""
category = "shape-independent"
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [["cor_length", "Ang", 50.0, [0, inf], "", "correlation length"],
]
Iq = """
double numerator = cube(cor_length);
double denominator = square(1 + square(q*cor_length));
return numerator / denominator ;
"""
def random():
"""Return a random parameter set for the model."""
pars = dict(
scale=10**np.random.uniform(1, 4),
cor_length=10**np.random.uniform(0.3, 3),
# background = 0,
)
pars['scale'] /= pars['cor_length']**3
return pars
demo = dict(scale=1, background=0, cor_length=50)
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