\end{equation}
By this means values of 2.973 nm and 4.309 \AA{} are obtained for the initial precipitate radius and lattice constant of 3C-SiC.
Since the generation of atoms is a discrete process with regard to the size of the volume the expected amounts of atoms are not obtained.
-However, by applying these values the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in table \ref{table:md:sic_prec}.
+However, by applying these values the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in table~\ref{table:md:sic_prec}.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{l c c c c}
\caption[Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$. The Si-Si radial distribution of plain c-Si is plotted for comparison. Green arrows mark bumps in the Si-Si distribution of the precipitate configuration, which do not exist in plain c-Si.}
\label{fig:md:pc_sic-prec}
\end{figure}
-Figure \ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
+Figure~\ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of 0.235 nm, which is the distance of next neighboured Si atoms in c-Si.
Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly there is no change at all within observational accuracy.
Looking closer at higher order Si-Si peaks might even allow the guess of a slight increase of the lattice constant compared to the plain c-Si structure.
\frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC of precipitate configuration}}}}
{\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}}
\end{equation}
-with the notation used in table \ref{table:md:sic_prec}.
+with the notation used in table~\ref{table:md:sic_prec}.
The lattice constant of plain c-Si at $20\,^{\circ}\mathrm{C}$ can be determined more accurately by the side lengthes of the simulation box of an equlibrated structure instead of using the radial distribution data.
By this a value of $a_{\text{plain c-Si}}=5.439\text{ \AA}$ is obtained.
The same lattice constant is assumed for the c-Si surrounding in the precipitate configuration $a_{\text{c-Si of precipitate configuration}}$ since peaks in the radial distribution match the ones of plain c-Si.
In the following the 3C-SiC/c-Si interface is described in further detail.
One important size analyzing the interface is the interfacial energy.
It is determined exactly in the same way than the formation energy as described in equation \eqref{eq:defects:ef2}.
-Using the notation of table \ref{table:md:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
+Using the notation of table~\ref{table:md:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
\begin{equation}
E_{\text{f}}=E-
N^{\text{3C-SiC}}_{\text{C}} \mu_{\text{SiC}}-
\caption{Free energy and temperature evolution of a constructed 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
\label{fig:md:fe_and_t_sic}
\end{figure}
-Figure \ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution.
+Figure~\ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution.
The sudden increase of the free energy indicates possible melting occuring around 2840 K.
\begin{figure}[!ht]
\begin{center}
\caption{Radial distribution of the constructed 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
\label{fig:md:pc_500-fin}
\end{figure}
-Investigating the radial distribution function shown in figure \ref{fig:md:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the free energy plot.
+Investigating the radial distribution function shown in figure~\ref{fig:md:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the free energy plot.
However the precipitate itself is not involved, as can be seen from the Si-C and C-C distribution, which essentially stays the same for both temperatures.
Thus, it is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two Si-Si distributions.
-This is surprising since the melting transition of plain c-Si is expected at temperatures around 3125 K, as discussed in section \ref{subsection:md:tval}.
+This is surprising since the melting transition of plain c-Si is expected at temperatures around 3125 K, as discussed in section~\ref{subsection:md:tval}.
Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
This is indeed verified by visualizing the atomic data.
% ./visualize -w 640 -h 480 -d saves/sic_prec_120Tm_cnt1 -nll -11.56 -0.56 -11.56 -fur 11.56 0.56 11.56 -c -0.2 -24.0 0.6 -L 0 0 0.2 -r 0.6 -B 0.1
\caption{Cross section image of atomic data gained by annealing simulations of the constructed 3C-SiC precipitate in c-Si at 200 ps (top left), 520 ps (top right) and 720 ps (bottom).}
\label{fig:md:sic_melt}
\end{figure}
-Figure \ref{fig:md:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
+Figure~\ref{fig:md:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
As can be seen from the image at 520 ps melting of the Si surrounding in fact starts in the defective interface region of the 3C-SiC precipitate and the c-Si surrounding propagating outwards until the whole Si matrix is affected at 720 ps.
As predicted from the radial distribution data the precipitate itself remains stable.
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