\subsection{Potential and exchange-correlation functional}
To find the most suitable combination of potential and XC functional for the C/Si system a $2\times2\times2$ supercell of type 3 of Si and C, both in the diamond structure, as well as 3C-SiC is equilibrated for different combinations of the available potentials and XC functionals.
-To exclude a possibly corrupting influence of the other parameters highly accurate calculations are performed, i.e. an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $k$-point mesh is used.
+To exclude a possibly corrupting influence of the other parameters highly accurate calculations are performed, i.e.\ an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $k$-point mesh is used.
Next to the ultra-soft pseudopotentials \cite{vanderbilt90} \textsc{vasp} offers the projector augmented-wave method (PAW) \cite{bloechl94} to describe the ion-electron interaction.
The two XC functionals included in the test are of the LDA \cite{ceperley80,perdew81} and GGA \cite{perdew86,perdew92} type as implemented in \textsc{vasp}.
Table \ref{table:simulation:potxc} shows the lattice constants and cohesive energies obtained for the fully relaxed structures with respect to the utilized potential and XC functional.
As expected, cohesive energies are poorly reproduced by the LDA whereas the equilibrium lattice constants are in good agreement.
Using GGA together with the ultra-soft pseudopotential yields improved lattice constants and, more importantly, a very nice agreement of the cohesive energies to the experimental data.
-The 3C-SiC calculations employing the PAW method in conjunction with the LDA suffers from the general problem inherent to LDA, i.e. overestimated binding energies.
+The 3C-SiC calculations employing the PAW method in conjunction with the LDA suffers from the general problem inherent to LDA, i.e.\ overestimated binding energies.
Thus, the PAW \& LDA combination is not pursued.
Since the lattice constant and cohesive energy of 3C-SiC calculated by the PAW method using the GGA are not improved compared to the ultra-soft pseudopotential calculations using the same XC functional, this concept is likewise stopped.
To conclude, the combination of ultra-soft pseudopotentials and the GGA XC functional are considered the optimal choice for the present study.
The quality of the integration algorithm and the occupied time step is determined by the ability to conserve the total energy.
Therefore, simulations of a $9\times9\times9$ 3C-SiC unit cell containing 5832 atoms in total are carried out in the $NVE$ ensemble.
-The calculations are performed for \unit[100]{ps} corresponding to $10^5$ integration steps and two different initial temperatures are considered, i.e. \unit[0]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C}.
+The calculations are performed for \unit[100]{ps} corresponding to $10^5$ integration steps and two different initial temperatures are considered, i.e.\ \unit[0]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{verlet_e.ps}
%Since the precipitate configuration is artificially constructed, the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate.
%Thus, annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
%The precipitate structure is rapidly heated up to \unit[2050]{$^{\circ}$C} with a heating rate of approximately \unit[75]{$^{\circ}$C/ps}.
-%From that point on the heating rate is reduced to \unit[1]{$^{\circ}$C/ps} and heating is continued upto \unit[120]{\%} of the Si melting temperature of the potential, i.e. \unit[2940]{K}.
+%From that point on the heating rate is reduced to \unit[1]{$^{\circ}$C/ps} and heating is continued upto \unit[120]{\%} of the Si melting temperature of the potential, i.e.\ \unit[2940]{K}.
%\begin{figure}[t]
%\begin{center}
%\includegraphics[width=0.7\textwidth]{fe_and_t_sic.ps}
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