+For the Monte Carlo simulation the target is devided into cells with a side length of $3 \, nm$.
+Each cell has a crystalline or amorphous state and stores the local carbon concentration.
+It is addressed by a position vector $\vec{r} = (k,l,m)$ where $k$, $l$ and $m$ are integers.
+The simulation starts with a complete crytsalline target and zero carbon inside.
+
+The model proposes three mechanisms of amorphization.
+In the simulation, each of this mechanisms contributes to a local amorphization probability of cell $\vec{r}$.
+The influence of the mechanisms are controlled by simulation parameters.
+The local amorphization probability at volume $\vec{r}$ is calculated by
+\begin{equation}
+p_{c \rightarrow a}(\vec{r}) = p_b + p_c c_C(\vec{r}) + \sum_{\textrm{amorphous neighbours}} \frac{p_s c_C(\vec{r'})}{(r-r')^2} \textrm{ .}
+\end{equation}
+
+The ballistic amorphization is constant and controlled by $p_b$.
+This choice is justified by analysing {\em TRIM} \cite{trim} collision data that show a mean constant energy loss per collision of an ion.
+The carbon induced amorphization is proportional to the local amount of carbon $c_C(\vec{r})$ and controlled by the simulation parameter $p_c$.
+The stress enhanced amorphization is controlled by $p_s$.
+The forces originating from the amorphous volumes $\vec{r}'$ in the vicinity are assumed to be proportional to the amount of carbon $c_C(\vec{r}')$.
+The sum is just taken over volumes located in the layer and since the stress amplitude is decreasing with the square of the distance $r-r'$ a cutoff radius is used in the simulation.
+In case of an amorphous volume, a recrystallization probability is given by
+\begin{equation}
+p_{a \rightarrow c}(\vec r) = (1 - p_{c \rightarrow a}(\vec r)) \Big(1 - \frac{\sum_{direct \, neighbours} \delta (\vec{r'})}{6} \Big) \, \textrm{,}
+\end{equation}
+\[
+\delta (\vec r) = \left\{
+\begin{array}{ll}
+ 1 & \textrm{volume at position $\vec r$ amorphous} \\
+ 0 & \textrm{otherwise} \\
+\end{array}
+\right.
+\]
+which is basically $1$ minus the amorphization probability and a term taking into account the crystalline neighbourhood which is needed for epitaxial recrystallization.
+
+The simulation algorithm consists of three parts.
+The first part is the amorphization/recrystallization step.
+Random values are computed to specify the volume $\vec{r}$ which is hit by an impinging carbon ion.
+Two uniformly distributed random numbers $x$ and $y$ are mapped to the coordinates $k$ and $l$.
+A random number $z$ corresponding to the depth coordinate $m$ is distributed according to the nuclear stopping power gained by {\em TRIM} using the rejection method.
+
+Again analysing {\em TRIM} collision data shows the nuclear stopping power being essentially
+The amorphization or recrystallization probability is computed and another random number decides whether there is amorphization or recrystallization or
+
+