Obviously, inverting a matrix at every timestep in the simulation of temperature is computationally prohibitive, but the calculations presented here are carried out at 0K, therefore making the latter method more effective since the dynamical matrix has already been calculated. The final method for a crystal containing many atoms in a single primitive unit cell will be given here. The elastic coefficients can be calculated directly from as follows. Therefore will be singular since it will have linearly dependant rows and columns. The atomistic relaxation simulation which relaxed the atoms into the lowest local energy configuration and the periodic boundary conditions on the supercells in conjunction with the Parrinello-Rahman Lagrangian ensures that these conditions are satisfied.
Table shows the independent elastic coefficients for the various supercells in the two simulations.
The entire elastic matrix is calculated in all cases. It is found that , and . All other coefficients are almost zero . It can be seen, however, that the change in bulk modulus seems to be almost independent of the type of defect under consideration. The elastic coefficients are also calculated for the smaller supercells, which allows the change in elastic coefficients with respect to oncentration to be calculated.