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 (\(8\times8\times8\) conventional unit cells).

Todo

Insert proper reference via references.bib.

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

\[ \begin{align}\begin{aligned}S_{NN}(q,\omega) = S_{AA}(q,\omega) + S_{BB}(q,\omega) + S_{AB}(q,\omega)\\S_{MM}(q,\omega) = ( m_A^2 S_{AA}(q,\omega) + m_B^2 S_{BB}(q,\omega) + m_A m_B S_{AB}(q,\omega) ) / (m_A+m_B)^2\\S_{ZZ}(q,\omega) = q_A^2 S_{AA}(q,\omega) + q_B^2 S_{BB}(q,\omega) + q_Aq_B S_{AB}(q,\omega),\end{aligned}\end{align} \]

where \(m_A\) and \(q_A\) denote the mass and charge of atom type \(A\), 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.

Structure factors: F(q,t) and S(q,w)

The figure below shows the static structure factors.

../_images/structureFactor.png

Number-number and charge-charge structure factor (left) and partial structure factors (right).

All structure factors agree well with Figures 10.2 and 10.3 in [JPH90]. The normalization differs by a factor of two, which is simply a matter of choice if the normalization for the partial structure factor should be done with \(N_A\) or \(N_A+N_B\).

Below the partial intermediate scattering functions and dynamical structure factors are shown.

../_images/S_partial.png

Partial \(F(q,t)\) and \(S(q,\omega)\) for two different q-values.

../_images/S_CMN.png

Mass charge and number computed from partial \(F(q,t)\) and \(S(q,\omega)\). Note the logged scale for \(S(q,\omega)\) in order to resolve some peaks.

../_images/S_partial_map.png

Map of logged partial dynamical structure factors.

../_images/S_CMN_map.png

Map of logged charge, mass and number dynamical structure factors.

Charge correlation: C(q,t) and C(q,w)

Below the partial, charge, mass and number longitudinal and transverse current correlation functions are shown. Note that the peaks in the frequency domain are much more easily seen compared to the dynamical structure factors. These figures show four modes (TA, TO, LA, LO).

../_images/C_partial.png

Partial current correlations \(C(q,t)\) and \(C(q,\omega)\).

../_images/C_CMN.png

Mass, charge and number current correlations \(C(q,t)\) and \(C(q,\omega)\).

Below maps of the current correlations are shown.

../_images/C_partial_map.png

Map of partial current correlations.

../_images/C_CMN_map.png

Map of current, mass and number current correlations.