$V_1$ is chosen to be the total simulation volume.
$V_2$ approximately corresponds to the volume of a minimal 3C-SiC precipitate.
$V_3$ is approximately the volume containing the necessary amount of silicon atoms to form such a precipitate, which is slightly smaller than $V_2$ due to the slightly lower silicon density of 3C-SiC compared to c-Si.
-For rectangularly shaped precipitates with side length $L$ equation \eqref{eq:md:quadratic_prec} holds.
+For rectangularly shaped precipitates with side length $L$ the amount of carbon atoms in 3C-SiC and silicon atoms in c-Si is given by
\begin{equation}
- N_{\text{Carbon}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
+ N_{\text{Carbon}}^{\text{3C-SiC}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
\label{eq:md:quadratic_prec}
\end{equation}
-Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by the equations mentioned above.
+and
+\begin{equation}
+ N_{\text{Silicon}}^{\text{c-Si}} =8 \left( \frac{L}{a_{\text{Si}}}\right)^3 \text{ .}
+\label{eq:md:quadratic_prec2}
+\end{equation}
+Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by equations \eqref{eq:md:quadratic_prec} and \eqref{eq:md:quadratic_prec2} and the resulting carbon concentrations inside these volumes.
\begin{table}
\begin{center}
\begin{tabular}{l c c c}
& $V_1$ & $V_2$ & $V_3$ \\
\hline
Side length [\AA] & 168.3 & 50.0 & 49.0 \\
+Carbon concentration [$\frac{1}{\text{c-Si unit cell}}$] & 0.20 & 7.68 & 8.16\\
\hline
\hline
\end{tabular}
Thus, the simulation is continued without adding more carbon atoms until the system temperature is equal to the chosen temperature again, which is realized by the thermostat decoupling excessive energy.
Every inserted carbon atom must exhibit a distance greater or equal than 1.5 \AA{} to present neighboured atoms to prevent too high temperatures.
Once the total amount of carbon is inserted the simulation is continued for 100 ps followed by a cooling-down process until room temperature, that is $20\, ^{\circ}\mathrm{C}$ is reached.
-Figure \ref{} displays a flow chart of the applied steps involved in the simulation sequence.
+Figure \ref{fig:md:prec_fc} displays a flow chart of the applied steps involved in the simulation sequence.
+\begin{figure}
+\begin{center}
+\begin{pspicture}(0,0)(15,17)
+
+ \psframe*[linecolor=hb](3,11.5)(11,17)
+ \rput[lt](3.2,16.8){\color{gray}INITIALIZIATION}
+ \rput(7,16){\rnode{14}{\psframebox{Create $31\times 31\times 31$
+ unit cells of c-Si}}}
+ \rput(7,15){\rnode{13}{\psframebox{$T_{\text{s}}=450\,^{\circ}\mathrm{C}$,
+ $p_{\text{s}}=0\text{ bar}$}}}
+ \rput(7,14){\rnode{12}{\psframebox{Thermal initialization}}}
+ \rput(7,13){\rnode{11}{\psframebox{Continue for 100 fs}}}
+ \rput(7,12){\rnode{10}{\psframebox{$T_{\text{avg}}=T_{\text{s}}
+ \pm1\,^{\circ}\mathrm{C}$}}}
+ \ncline[]{->}{14}{13}
+ \ncline[]{->}{13}{12}
+ \ncline[]{->}{12}{11}
+ \ncline[]{->}{11}{10}
+ \ncbar[angle=0]{->}{10}{11}
+ \psset{fillcolor=hb}
+ \nbput*{\scriptsize false}
+
+ \psframe*[linecolor=lbb](3,6.5)(11,11)
+ \rput[lt](3.2,10.8){\color{gray}CARBON INSERTION}
+ \rput(3,10.8){\pnode{CI}}
+ \rput(7,10){\rnode{9}{\psframebox{Insertion of 10 carbon aoms}}}
+ \rput(7,9){\rnode{8}{\psframebox{Continue for 100 fs}}}
+ \rput(7,8){\rnode{7}{\psframebox{$T_{\text{avg}}=T_{\text{s}}
+ \pm1\,^{\circ}\mathrm{C}$}}}
+ \rput(7,7){\rnode{6}{\psframebox{$N_{\text{Carbon}}=6000$}}}
+ \ncline[]{->}{9}{8}
+ \ncline[]{->}{8}{7}
+ \ncline[]{->}{7}{6}
+ \trput*{\scriptsize true}
+ \ncbar[angle=180]{->}{7}{8}
+ \psset{fillcolor=lbb}
+ \naput*{\scriptsize false}
+ \ncbar[angle=0]{->}{6}{9}
+ \nbput*{\scriptsize false}
+ \ncbar[angle=180]{->}{10}{CI}
+ \psset{fillcolor=white}
+ \nbput*{\scriptsize true}
+
+ \rput(7,5.75){\rnode{5}{\psframebox{Continue for 100 ps}}}
+ \ncline[]{->}{6}{5}
+ \trput*{\scriptsize true}
+
+ \psframe*[linecolor=lachs](3,0.5)(11,5)
+ \rput[lt](3.2,4.8){\color{gray}COOLING DOWN}
+ \rput(3,4.8){\pnode{CD}}
+ \rput(7,4){\rnode{4}{\psframebox{$T_{\text{s}}=T_{\text{s}}-
+ 1\,^{\circ}\mathrm{C}$}}}
+ \rput(7,3){\rnode{3}{\psframebox{Continue for 100 fs}}}
+ \rput(7,2){\rnode{2}{\psframebox{$T_{\text{avg}}=T_{\text{s}}
+ \pm1\,^{\circ}\mathrm{C}$}}}
+ \rput(7,1){\rnode{1}{\psframebox{$T_{\text{s}}=20\,^{\circ}\mathrm{C}$}}}
+ \ncline[]{->}{4}{3}
+ \ncline[]{->}{3}{2}
+ \ncline[]{->}{2}{1}
+ \trput*{\scriptsize true}
+ \ncbar[angle=0]{->}{2}{3}
+ \psset{fillcolor=lachs}
+ \nbput*{\scriptsize false}
+ \ncbar[angle=180,arm=1.5]{->}{1}{4}
+ \naput*{\scriptsize false}
+ \ncbar[angle=180]{->}{5}{CD}
+ \trput*{\scriptsize false}
-The radial distribution function for Si-C and C-C distances is shown in figure \ref{}.
+ \rput(7,-0.25){\rnode{0}{\psframebox{End of simulation}}}
+ \ncline[]{->}{1}{0}
+ \trput*{\scriptsize true}
+\end{pspicture}
+\end{center}
+\caption{Flowchart of the simulation sequence used in the molecular dnymaics simulations aiming to reproduce the precipitation process.}
+\label{fig:md:prec_fc}
+\end{figure}
+
+The radial distribution function $g(r)$ for Si-C and C-C distances is shown in figure \ref{fig:md:pc_si-si_c-c}.
+\begin{figure}[!ht]
+\begin{center}
+ \includegraphics[width=8cm]{pc_si-c_c-c_thesis.ps}
+\end{center}
+\caption{Radial distribution function of the Si-C and C-C distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$.}
+\label{fig:md:pc_si-si_c-c}
+\end{figure}
\subsection{Increased temperature simulations}
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