# Surface Ripple Formation by Bombardment with Clusters: Influence of Mass
## 2 Model
The equations of motion of the particle system are derived from the Lagrangian of the so-called extended system, where the two previous adjustable parameters can appear depending on the ensemble. The last step consists of eliminating the periodic conditions in the \(z\) direction and relaxing the surface into a microcanonical ensemble coupled to the thermal bath. The process of launching a cluster lasted 25 ps. This time interval is similar to that used in other works [20]. At first, the time step was 0.5 fs and 0.1 fs in the case of He. After 15 ps from launch, this time step was doubled, since the collisions were less energetic. At 20 ps, the system was scaled to 300 K. It was then relaxed for 5 ps so that the temperature conditions were the same on the next launch. The evolution of the system took place at all times in the microcanonical ensemble coupled to the thermal bath. For simplicity, in each launch, the reference frame was translated in the \(y\) direction so that the cluster always hit the center of the surface. Of course, the model used does not depend on the \(y\) coordinate. The surface was bombarded with 167 clusters, 1002 atoms in total with 250 eV each. The \(x\) coordinate of the geometric center of the cluster initially points to a point randomly located between \(-10\) and \(0\) A. In this way, the bombardment occurred on a strip parallel to the \(Oy\) axis. The number of incident atoms per unit area (fluence) is approximately \(10^{16}\) atoms-cm\({}^{-2}\). Experimentally, atom bombardment on Co at higher energies and slightly higher fluences produces surface ripples [21]. In our results, the fluence is not as high; however, the effect is amplified by cluster bombardment. There are two main effects that alter the surface morphology. One of them is the formation of craters. Craters occur as a result of the extraction of atoms from the solid. In our simulations, each atom in the cluster extracts a number of solid atoms. This number \(Y\), or sputter yield, can be obtained. The other is the formation of surface structures. Solid atoms move and form vacancies and surface accumulations as a result of induced atomic collisions with the cluster atoms. In general, this process of redistribution of atoms is measured by two displacements [8]: the horizontal displacement \(d_{x}\) and the vertical displacement \(d_{z}\). These quantities measure the net distance traveled by the atoms of the solid in the two directions of initial motion of the cluster, \(x\) and \(z\). \[d_{x}=\sum_{i=1}^{N_{d}}(x_{i}-x_{i0}),\;d_{z}=\sum_{i=1}^{N_{d}}(z_{i}-z_{i0}). \tag{2}\]  
The previous sum extends to the number of displaced atoms \(N_{d}\). In turn, \((x_{i0},y_{i0},z_{i0})\) and \((x_{i},y_{i},z_{i})\) are the initial and generic positions of any atom \(i\) of the substrate, respectively. If \(d_{x}\) is positive, there is mass transport in the positive \(x\) direction. If \(d_{z}\) is positive, there is upward mass transport opposite to the initial movement of the cluster atoms. This work analyzes the influence of the angle of incidence on the previous parameters for clusters formed by atoms of different atomic masses. This angle of incidence varied between 20\({}^{\circ}\) and 70\({}^{\circ}\).