-The assumed applicability as discussed above and the remaining absence of either agglomeration of substitutional C in low concentration simulations or amorphous to crystalline transition in high concentration simulations suggests to further increase the system temperature.
-So far, the highest temperature applied corresponds to 95 \% of the absolute silicon melting temperature, which is 2450 K and specific to the Erhard/Albe potential.
-However, melting is not predicted to occur instantly after exceeding the melting point due to additionally required transition enthalpy and hysteresis behaviour.
-To check for the possibly highest temperature at which a transition fails to appear plain silicon is heated up using a heating rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
-Figure \ref{fig:md:fe_and_t} shows the free energy and temperature evolution in the region around the transition temperature.
-Indeed a transition and the accompanying critical behaviour of the free energy is first observed at approximately 3125 K, which corresponds to 128 \% of the silicon melting temperature.
-The difference in free energy is 0.58 eV per atom corresponding to $55.7 \text{ kJ/mole}$, which compares quite well to the silicon enthalpy of melting of $50.2 \text{ kJ/mole}$.
-The late transition probably occurs due to the high heating rate and, thus, a large hysteresis behaviour extending the temperature of transition.
-To avoid melting transitions in further simulations system temperatures well below the transition point are considered safe.
-Thus, in the following system temperatures of 100 \% and 120 \% of the silicon melting point are used.
-
-\subsection{Constructed 3C-SiC precipitate in crystalline silicon}
-
-{\color{red}Todo: We want to know where we want to go ...}
-
-In the following a spherical 3C-SiC precipitate enclosed in a c-Si surrounding is constructed as it is expected from IBS experiments and from simulations that finally succeed simulating the precipitation event.
-On the one hand this sheds light on characteristic values like the radial distribution function or the total amount of energy for configurations that are aimed to be reproduced by simulation possibly enabling the prediction of conditions necessary for the simulation of the precipitation process.
-On the other hand, assuming a correct alignment of the precipitate with the c-Si matrix, investigations of the behaviour of such precipitates and the surrounding can be made.
-
-To construct a spherical 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied.
-A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is used.
-To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
-The initial precipitate configuration is constructed in two steps.
-In the first step the surrounding silicon matrix is created.
-This is realized by just skipping the generation of silicon atoms inside a sphere of radius $x$, which is the first unknown variable.
-The silicon lattice constant $a_{\text{Si}}$ of the surrounding c-Si matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation.
-In a second step 3C-SiC is created inside the empty sphere of radius $x$.
-The lattice constant $y$, the second unknown variable, is chosen in such a way, that the necessary amount of carbon is generated.
-This is entirely described by the system of equations \eqref{eq:md:constr_sic_01}
-\begin{equation}
-\frac{8}{a_{\text{Si}}^3}(
-\underbrace{21^3 a_{\text{Si}}^3}_{=V}
--\frac{4}{3}\pi x^3)+
-\underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
-=21^3\cdot 8
-\label{eq:md:constr_sic_01}
-\text{ ,}
-\end{equation}
-which can be simplified to read
-\begin{equation}
-\frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
-\Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
-\label{eq:md:constr_sic_02}
-\end{equation}
-and
-\begin{equation}
-%x^3=\frac{16\pi}{5500 \cdot 3}y^3=
-%\frac{16\pi}{5500 \cdot 3}\frac{5500 \cdot 3}{32 \pi}a_{\text{Si}}^3
-%\Rightarrow
-y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
-\text{ .}
-\label{eq:md:constr_sic_03}
-\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}.
-\begin{table}[!ht]