\]
with $b_{ap}$, $a_{cp}$ and $a_{ap}$ being parameters of the simulation to weight the three different ways of amorphization. The probability of an amorphous volume turning crystalline should behave contrary to $p_{c \rightarrow a}$ and thus is assumed to $p_{a \rightarrow c} = 1 - p_{c \rightarrow a}$.
-Figure \ref{3} shows the target devided into $64 \times 64 \times 100$ volumes with a side length of $3 \, nm$. Each of it has a state (crystalline/amorphous) and keeps the local carbon concentration. The simulation algorithm consists of three parts, the amorphization/recrystallization process, the carbon-implanting process and finally the diffusion process.
+Figure \ref{3} shows the target devided into $64 \times 64 \times 100$ volumes with a side length of $3 \, nm$. Each of it has a state (crystalline/amorphous) and keeps the local carbon concentration. The simulation algorithm consists of three parts, the amorphization/recrystallization process, the carbon-incorporation process and finally the diffusion process.
For the amorphization/recrystallization process, random coordinates are computed to specify the volume which is hit by an implanted carbon ion. The two random numbers corresponding to the $x$ and $y$ coordinates are generated with a uniform probability distribution, $p(x)dx=dx \textrm{, } p(y)dy=dy$. The random number corresponding to the $z$ coordinate is distributed according to the linear approximated nuclear stopping power, $p(z)dz=(a_{el} \times z+b_{el})dz$, where $a_{el}$ and $b_{el}$ are simulation parameters describing the nuclear energy loss. After calculating the local probability of amorphization of that volume $p_{c \rightarrow a}$, another random number decides, depending on the current state, whether the volume gets amorphous or recrystallized. This step is looped for the average hit per ion in the simulation window, counted by TRIM collision data.
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