source: sasmodels/sasmodels/models/poly_gauss_coil.py @ 58c3367

#poly_gauss_coil model
#conversion of Poly_GaussCoil.py
#converted by Steve King, Mar 2016
r"""
This empirical model describes the scattering from *polydisperse* polymer
chains in theta solvents or polymer melts, assuming a Schulz-Zimm type
molecular weight distribution.

To describe the scattering from *monodisperse* polymer chains, see the
:ref:`mono-gauss-coil` model.

Definition
----------

.. math::

     I(q) = \text{scale} \cdot I_0 \cdot P(q) + \text{background}

where

.. math::

     I_0 &= \phi_\text{poly} \cdot V \cdot (\rho_\text{poly}-\rho_\text{solv})^2

     P(q) &= 2 [(1 + UZ)^{-1/U} + Z - 1] / [(1 + U) Z^2]

     Z &= [(q R_g)^2] / (1 + 2U)

     U &= (Mw / Mn) - 1 = \text{polydispersity ratio} - 1

     V &= M / (N_A \delta)

Here, $\phi_\text{poly}$, is the volume fraction of polymer, $V$ is the
volume of a polymer coil, $M$ is the molecular weight of the polymer,
$N_A$ is Avogadro's Number, $\delta$ is the bulk density of the polymer,
$\rho_\text{poly}$ is the sld of the polymer, $\rho_\text{solv}$ is the
sld of the solvent, and $R_g$ is the radius of gyration of the polymer coil.

The 2D scattering intensity is calculated in the same way as the 1D,
but where the $q$ vector is redefined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}

References
----------

O Glatter and O Kratky (editors), *Small Angle X-ray Scattering*,
Academic Press, (1982) Page 404.

J S Higgins, H C Benoit, *Polymers and Neutron Scattering*,
Oxford Science Publications, (1996).

S M King, *Small Angle Neutron Scattering* in *Modern Techniques for
Polymer Characterisation*, Wiley, (1999).

http://www.ncnr.nist.gov/staff/hammouda/distance_learning/chapter_28.pdf
"""

from numpy import inf, exp, power

name = "poly_gauss_coil"
title = "Scattering from polydisperse polymer coils"

description = """
    Evaluates the scattering from 
    polydisperse polymer chains.
    """
category = "shape-independent"

# pylint: disable=bad-whitespace, line-too-long
#   ["name", "units", default, [lower, upper], "type", "description"],
parameters = [
    ["i_zero",          "1/cm", 70.0, [0.0, inf], "", "Intensity at q=0"],
    ["radius_gyration",  "Ang", 75.0, [0.0, inf], "", "Radius of gyration"],
    ["polydispersity",  "None",  2.0, [1.0, inf], "", "Polymer Mw/Mn"],
    ]
# pylint: enable=bad-whitespace, line-too-long

# NB: Scale and Background are implicit parameters on every model
def Iq(q, i_zero, radius_gyration, polydispersity):
    # pylint: disable = missing-docstring
    u = polydispersity - 1.0
    z = (q*radius_gyration)**2 / (1.0 + 2.0*u)
    # need to trap the case of the polydispersity being 1 (ie, monodisperse!)
    if polydispersity == 1.0:
        inten = i_zero * 2.0 * (exp(-z) + z - 1.0)
    else:
        inten = i_zero * 2.0 * (power(1.0 + u*z, -1.0/u) + z - 1.0) / (1.0 + u)
    index = q != 0.
    inten[~index] = i_zero
    inten[index] /= z[index]**2
    return inten
Iq.vectorized = True  # Iq accepts an array of q values

demo = dict(scale=1.0,
            i_zero=70.0,
            radius_gyration=75.0,
            polydispersity=2.0,
            background=0.0)

# these unit test values taken from SasView 3.1.2
tests = [
    [{'scale': 1.0, 'i_zero': 70.0, 'radius_gyration': 75.0,
      'polydispersity': 2.0, 'background': 0.0},
     [0.0106939, 0.469418], [57.6405, 0.169016]],
    ]