The initial and boundary conditions

The initial configuration in our simulations is set up by a distribution of all atoms at the sites of a bcc lattice. Afterward the point defects are introduced either by insertion of extra atoms in a random fashion between the lattice sites (self-interstitials), or by removal of atoms from the lattice sites (vacancies). These point defects are located at long distances one from another to avoid, as far as it possible, their interactions. The initial velocities of all atoms and point defects are set to values which depend on the temperature according to the Maxwell distribution. When these random velocities have to be adjusted again to ensure that the center of mass of the system is stationary, (i.e the velocity of the center of mass is equal to zero).

Molecular dynamics is applied to systems containing usually a few thousands atoms. Surface effects, i.e. interactions of atoms with the container wall or effects of the free surface, are dominant in such small systems. In simulations of the crystal bulk those surface effects are not of interest and may be eliminated by means of periodic boundary conditions. In order to implement the periodic boundary conditions for $N$ atoms in a volume $V$, we imagine that the volume $V$ is only a small part of the bulk material. The volume $V$ is called the computational (or primary) cell, it represents the bulk material to extent that the bulk is assumed to be composed of the primary cell surrounded by exact replicas of itself. These replicas are called image cells. The image cells are the same size and shape. Thus, the primary cell is periodically replicated in all directions to form a macroscopic sample (See Fig. 3.1).

Figure: A computational cell and periodic boundary conditions

In order to simulate the surface the periodic boundary conditions have to be altered. We use periodic boundary conditions only in the $x$ and $y$ directions, which are parallel to the surface. The system is represented as a slab of layers oriented perpendicular to the $z$ direction. Each layer contains atoms arranged in a bcc pattern. The bottom layers are fixed to mimic the effects of an infinite bulk of the solid.