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4.4.1.1 Collision Partner in Amorphous Materials

Due to the fact the atoms in an amorphous material are approximately equally distributed a simple fast algorithm can be applied, because only the material density has to be considered by assuming that a cylinder with a radius of the maximum impact parameter $ p_{max}$ and a length of the average free flight path $ \overline{L}$ contains only one target atom. The average free flight path can then be calculated by

$\displaystyle p_{max}^2\cdot \pi\cdot \overline{L} = \frac{1}{\rho_N},$ (4.3)

if the particle density of the target material is $ \rho_N$. Since there is no difference in the simulation result no matter if the free flight path is kept constant or not [92], it makes sense to use a constant free flight path of $ \overline{L}$ . The collision partner can therefore be placed in a plane perpendicular to the direction of motion with a distance $ \overline{L}$ from the actual ion position (Fig. 4.4), while the probability for finding the collision partner within a circle with radius $ p_{max}$ in this plane is constant. The actual impact parameter $ p$ can therefore be calculated using a linear random number $ R_n$ in an interval of [0,1].

$\displaystyle p = p_{max} \cdot \sqrt{R_n}$ (4.4)

Figure 4.4: Schematic figure how the collision partner is determined in amorphous targets.
\begin{figure}\begin{center} \psfrag{pmax}{\large p$_\mathrm{max}$}\psfrag{p}{\l... ...tebox{0}{\includegraphics{fig/monte/AmorphPartner.eps}}}\end{center}\end{figure}

The maximum impact parameter is determined by the particle density.

$\displaystyle p_{max} = \frac{1}{\rho_N^3},$ (4.5)

Besides the position of the collision partner its material type has to be determined. The stoichiometry of the target material defines the probability for finding certain atom species in the material. Accordingly the type of the collision partner is selected randomly just considering this probability distribution. Thereby correlation effects due to the molecular structure in the material are neglected, but due to the large number of collisions all correlation effects are averaged.

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A. Hoessiger: Simulation of Ion Implantation for ULSI Technology