Method and Theory

First, a defect system is constructed by adding a single point defect (or defect cluster) to an initially bulk system using the atomman.defect.point() function.

A LAMMPS simulation is then performed on the defect system. The simulation consists of two separate runs

  1. NVT equilibrium run: The system is allowed to equilibrate at the given temperature using nvt integration.
  2. NVE measurement run: The system is then evolved using nve integration, and the total mean square displacement of all atoms is measured as a function of time.

Between the two runs, the atomic velocities are scaled such that the average temperature of the nve run is closer to the target temperature.

The mean square displacement of the defect, \(\left< \Delta r_{ptd}^2 \right>\) is then estimated using the mean square displacement of the atoms \(\left< \Delta r_{i}^2 \right>\). Under the assumption that all diffusion is associated with the single point defect, the defect’s mean square displacement can be taken as the summed square displacement of the atoms

\[\left< \Delta r_{ptd}^2 \right> \approx \sum_i^N \Delta r_{i}^2 = N \left< \Delta r_{i}^2 \right>,\]

where \(N\) is the number of atoms in the system. The diffusion constant is then estimated by linearly fitting the change in mean square displacement with time

\[\left< \Delta r_{ptd}^2 \right> = 2 d D_{ptd} \Delta t,\]

where d is the number of dimensions included.