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Equations of motion

Once the specific form of the potential is established, the forces between the atoms can be compute from the gradient of the potential. We are then lead to solve the differential equations :
   
$\displaystyle F_{i_\alpha} = m \ \ \frac{d^2r_{i_\alpha}}{dt^2} =
-\frac{\partial E_i} {\partial r_{i_\alpha}}$     (8.1)
       
$\displaystyle v_{i_\alpha} = \frac{dr_{i_\alpha}}{dt}$     (8.2)

in order to obtain the position ri and the velocity vi of each atom of mass m as a function of the time t. i is the atom in consideration and $\alpha$ the coordinates x, y and z. As explained before, in the case of the tight binding model, a force emerging from the electronic part of the total energy should also be calculated. However, like in the classical models, in the quantum or semi-classical approaches, the atoms are considered as classical, so that also in these cases, the Newton equations have to be solved.

The first step of the calculation consists of determining the neighbors of each atom within the limit of the force range. Then, for each atom the force applied by its neighbors is computed and added all together to get the total force on the atom. From the forces evaluation, the Newton's equations (8.1) and (8.2) are solved for coordinates x, y and z and velocities vx, vyand vz. For this purpose, two different algorithm were used in the present research and compared regarding the total energy obtained and the average atomic positions during the processes of bombardment and annealing.



 
next up previous contents
Next: The leapfrog algorithm Up: The numerical techniques Previous: The numerical techniques
David Saada
2000-06-22