source: sasmodels/sasmodels/models/mono_gauss_coil.py @ 38d2c97

#mono_gauss_coil model
#conversion of DebyeModel.py
#converted by Steve King, Mar 2016



r"""
This model strictly describes the scattering from *monodisperse* polymer chains in theta solvents or polymer melts, conditions under which the distances between segments follow a Gaussian distribution. Provided the number of segments is large (ie, high molecular weight polymers) the single-chain Form Factor P(Q) is that described by Debye (1947).

To describe the scattering from *polydisperse* polymer chains, see the poly_gauss_coil model.

Definition
----------

     *I(q)* = *scale* x *P(q)* + *background*
         
where

     *scale* = |phi|\ :sub:`poly` x *V* x (|rho|\ :sub:`poly` - |rho|\ :sub:`solv`)\ :sup:'2'

     *P(q)* = 2 [exp(-Z) + Z - 1] / Z\ :sup:'2'
         
         *Z* = (*q R*\ :sub:'g')\ :sup:'2'

and

         *V* = *M* / (*N*\ :sub:'A' |delta|)
         
Here, |phi|\ :sub:`poly`, is the volume fraction of polymer, *V* is the volume of a polymer coil, *M* is the molecular weight of the polymer, *N*\ :sub:'A' is Avogadro's Number, |delta| is the bulk density of the polymer, |rho|\ :sub:`poly` is the sld of the polymer, |rho|\ :sub:`solv` is the sld of the solvent, and *R*\ :sub:'g' is the radius of gyration of the polymer coil.

.. figure:: img/mono_gauss_coil_1d.jpg

    1D plot using the default values.

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

.. image:: img/2d_q_vector.gif

References
----------

P Debye, *J. Phys. Colloid. Chem.*, 51 (1947) 18.

R J Roe, *Methods of X-Ray and Neutron Scattering in Polymer Science*, Oxford University Press, New York (2000).

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

from numpy import inf, sqrt, exp

name =  "mono_gauss_coil"
title =  "Scattering from monodisperse polymer coils"

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

#             ["name", "units", default, [lower, upper], "type", "description"],
parameters =  [["radius_gyration", "Ang", 50.0, [0.0, inf], "", "Radius of gyration"]]

# NB: Scale and Background are implicit parameters on every model
def Iq(q, radius_gyration):
    # pylint: disable = missing-docstring
    z = (x * radius_gyration) * (x * radius_gyration)
    if x == 0:
       inten = 1.0
    else:
       inten = 2.0 * (exp(-z) + z - 1.0 ) / (z * z)
    return inten
Iq.vectorized =  True  # Iq accepts an array of q values

def Iqxy(qx, qy, *args):
    # pylint: disable = missing-docstring
    return Iq(sqrt(qx ** 2 + qy ** 2), *args)
Iqxy.vectorized =  True # Iqxy accepts an array of qx, qy values

demo =  dict(scale = 1.0,
            radius_gyration = 50.0,
            background = 0.0)

oldname =  "DebyeModel"
oldpars =  dict(scale = 'scale',
               radius_gyration = 'rg',
               background = 'background')

tests =  [
    [{'scale': 1.0, 'radius_gyration': 50.0, 'background': 0.0},
     [0.0106939, 0.469418], [0.911141, 0.00362394]],
    ]