Coordination number

As explained in chapter two, amorphous carbon solids ($ta-C$ as well as $a-C$) are characterized by the $sp^3/sp^2$ ratio between fourfold atoms $sp^3$ and treefold atoms $sp^2$. Each $sp^3$ bonded atom has four nearest neighbors separated by a distance of approximately 1.54 Å. Each $sp^2$ bonded atom has only three nearest neighbors separated by a shorter distance. Thus the method of distinguishing between $sp^3$ and $sp^2$ sites used in this work is based on determination of the coordination number of each atom.

In order to define the coordination number, we have calculated the radial distribution function $g(r)$ of the structures of amorphous carbon created after cooling of a liquid phase. We will return to a detailed discussion of $g(r)$ in the next chapter. Independently of density and cooling rate, $g(r)$ of all the samples exhibits a clear gap, centered at about 1.9 Å, separating the first and the second peak. All atoms within the sphere of radius 1.9 Å are thus assumed to comprise the first nearest neighborhood of a given atom. Therefore, the number of neighbors of each atom within a distance of 1.9 Å determines the coordination number.

In order to count the coordination number and to calculate the distances between all pairs of atoms for $g(r)$ the following procedure of determining of neighbors was used. An integer number (from 1 to 144 for the largest sample) is assigned to each atom that will permit identification of all atoms whenever needed. Then, for each atom number, $i$, its distance $r_{ij}$ to atom number $j$ is calculated, for $j$ running over all the atoms of the crystal. If $r_{ij}$ is lower than the distance of 1.9 Å, the label of the atom $j$ is stored in the list of nearest neighbors of the atom $i$. In order to calculate $g(r)$ the distances $r_{ij}$ are accumulated in a separate file. Afterwards, the number of the bond lengths restricted to the inteval from $r$ to $r+dr$ was summed up and divided by the number of atoms in the system.

The file of the nearest neighbors was also used for calculating an angular distribution function $g(\theta )$. Here, the angles between the atom $i$ and each pair of its nearest neigbors are accumulated in the list of angles.