.. _mono-gauss-coil:

mono_gauss_coil
=======================================================

Scattering from monodisperse polymer coils

========== =============================== ======= =============
Parameter  Description                     Units   Default value
========== =============================== ======= =============
scale      Scale factor or Volume fraction None                1
background Source background               |cm^-1|         0.001
i_zero     Intensity at q=0                |cm^-1|            70
rg         Radius of gyration              |Ang|              75
========== =============================== ======= =============

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


This Debye Gaussian coil 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
:ref:`poly-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 [\exp(-Z) + Z - 1] / Z^2 \\
     Z &= (q R_g)^2 \\
     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}


.. figure:: img/mono_gauss_coil_autogenfig.png

    1D plot corresponding to the default parameters of the model.


**Source**

:download:`mono_gauss_coil.py <src/mono_gauss_coil.py>`
$\ \star\ $ :download:`mono_gauss_coil.c <src/mono_gauss_coil.c>`

**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

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