sodium_chloride.rst 2.52 KB
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.. _example_sodium_chloride:

Molten Sodium Chloride
**********************

This example deals with molten sodium chloride in order to illustrate
the analysis of multiple component systems and the treatment of
partial correlation functions. As in the case of liquid alumnium the
analsysi is carried out for a spherically averaged momentum vector
**q** The `born/coul` potential in `lammps` was used with parameters
from Lewis and Singer (1974). The melting point was found to be about
1300 K and the simulations were carried out at 1400 K for 4096 atoms
(:math:`8\times8\times8` conventional unit cells).


The partial correlation functions are denoted :math:`S_{AA}(q,\omega)`
where :math:`A` is an atomic species and :math:`S` represents an
arbitray correlation function. We define the number correlation
:math:`S_{NN}(q,\omega)`, mass correlation :math:`S_{MM}(q,\omega)`
and the charge correlation :math:`S_{ZZ}(q,\omega)` as

.. math::

   S_{NN}(q,\omega) = S_{AA}(q,\omega) + S_{BB}(q,\omega) +
   S_{AB}(q,\omega)

   S_{MM}(q,\omega) = ( m_A^2 S_{AA}(q,\omega) + m_B^2
   S_{BB}(q,\omega) + m_A m_B S_{AB}(q,\omega) ) / (m_A+m_B)^2
   
   S_{ZZ}(q,\omega) = q_A^2 S_{AA}(q,\omega) + q_B^2
   S_{BB}(q,\omega) + q_Aq_B S_{AB}(q,\omega),

where :math:`m_A` and :math:`q_A` denote the mass and charge of atom
type :math:`A`, respectively. These are useful in ionized systems
because acoustic type modes show up in the mass correlation and optic
type modes can be seen in the charge correlation. For direct
comparison with with exprimental data one can compute a linear
combination of the partial functions weighted with the appropriate
atomic form factors.


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Structure factors: F(q,t)
=========================
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The figure below shows the static charge and mass structure factors.
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.. figure:: figs/NaCl_liquid/NaCl_Sq.png
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    :align: center
    
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    the static charge and mass structure factors.
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All structure factors agree well with Figures 10.2 and 10.3 in
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:cite:`HansenMcDonald`.
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Current correlations: C(q,w)
============================
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Below the partial, charge, mass and number longitudinal and transverse
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current correlation functions are shown.
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.. figure:: figs/NaCl_liquid/NaCl_partial_heatmaps.png
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    :align: center
    
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    Heatmap of partial current correlations. Top is longitudinal and bottom
    is transverse.
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.. figure:: figs/NaCl_liquid/NaCl_charge_mass_heatmaps.png
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    :align: center
    
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    Heatmap of charge and mass current correlations. Top is longitudinal and
    bottom is transverse.