Analysis of errors

The possible errors that can influence the results in computer simulation are: statistical errors, finite-size effects, unreliable generator of random numbers, unaccurate potential and numerical techniques, insufficient time of simulation to reach an equilibrium state.

The main source of systematic errors is the interatomic potential, which is in our case a tightbinding potential. It was shown in many previous simulations [1,15] that this potential is very reliable and gives an adequate description of amorphous phases of carbon. However, Marks et al. [74] found out that the tightbinding method yield a slightly lower $sp^3$ fraction at high densities in comparison with empirical potentials and the presence of singly coordinated atoms at low densities.

The predictor-corrector technique is sufficiently accurate. The generator of random numbers which is used in the present simulations was checked earlier to be reliable for this type of simulation [5,77]. So the contribution of these two factors to the total error is negligible.

Early attempts to prepare samples of amorphous carbon by quenching from the liquid carbon, when the liquid was not in an equilibrium state led to erroneous results. The analysis of these samples did not give any reliable results, the obtained data were scattered and statistical analysis could not help to reveal any regularity of the observed data. So in order to obtain adequate results it is very important to reach equilibrium and in all subsequent simulations the approach to equilibrium in the liquid phase was controlled thoroughly by monitoring of the total energy of the system.

In order to gather a set of better statistics, we should repeat our simulation with another initial conditions (initial velocities, seeds of random number generator). However, computer simulations based on the tightbinding method are very time and memory consuming, so we repeat our simulations only three times. Thus the statistical error of these calculations gives a large contribution to the total error. The three simulations gave similar results (in contrast to the mess when equilibrium was not achieved in preliminary simulations). The average statistical error is calculated according to:

$$\Delta \overline x =\frac{s}{\sqrt{k}}$$ (7.11)

where $s$ is a standard deviation:

$$s=\sqrt{\frac{\sum\limits_{i=1}^k (x_i -\overline x)^2}{k-1}}$$ (7.12)

and $\overline x$ is a simple arithmetic mean value:

$$\overline x = \frac{1}{k} \sum_{i=1}^k x_{i}.$$ (7.13)

Finally finite size effects are investigated systematically in this work. An amorphous carbon samples of different sizes prepared under similar conditions are compared. The results of this comparison are discussed.