# Surface Ripple Formation by Bombardment with Clusters: Influence of Mass
## 2 Model  
In this study, the bombardment is focused on a longitudinal strip parallel to the \(y\) axis, similar to a linearly focused ion beam (Figure 1). This axis is parallel to the crystallographic <001> direction. This approach reduces the time it takes for the surface response to occur. This model has been previously introduced to analyze this phenomenon in other works [8; 9]. Its main advantage is that it is simple and allows us to obtain results and draw conclusions from the processes involved. Furthermore, certain quantities, such as volumes, can be measured with some ease. A bombardment with a nonlinear focus would be a combination of the processes analyzed here. In fact, it is to be expected that the bombardment situation described here will also experimentally lead to pattern formation, with which our results will be completely valid for any bombardment process. The bombarded surface is a face-centered cubic (FCC) Co surface oriented in the <110> direction. The surface has dimensions of 56 \(\times\) 28 \(\times\) 14 unit cells (14.0 \(\times\) 9.9 \(\times\) 3.4 nm). The solid is considered to be periodic in the \(x\) and \(y\) directions (Figure 1). The three lowest \(z\)-layers remained fixed, and the five layers above these thermally controlled the system like a thermal bath. Their atoms satisfy a generalized [PERSON] equation in which the Debye temperature of FCC Co (385 K) is introduced as a parameter [11]. In essence, the force acting on an atom is written as the sum of a viscous force proportional to its speed and a random term that represents the effect of collisions with surrounding atoms. The interaction between cobalt atoms took place with a second-moment tight binding potential [12]. Experimentally, this potential fits the lattice constant, cohesive energy, and elastic constants for Co. In this model, the cohesive energy \(E_{i}\) of an atom \(i\) is:  
\[E_{i}=\sum_{j(\
eq i)}\frac{1}{2}V(r_{ij})-(\sum_{j(\
eq i)}\Phi)^{1/2}. \tag{1}\]  
Figure 1: Ar cluster and substrate before bombardment. The fixed atom zone appears in blue, the thermalization zone in red, and the bombarded surface in pink. An arrow marks the direction of movement of the cluster. where \(r_{ij}\) is the distance between atoms \(i\) and \(j\). The first term is repulsive in nature and is of the modified Born-Mayer type. The second term is of the attractive type and models the many-body interaction that constitutes the energy band. At short distances, this potential merged smoothly into a more repulsive one, such as the ZBL [13], which is used to describe high-energy interactions. This potential is basically a Coulomb potential multiplied by a screening function. It also modeled the interactions between cluster atoms and Co. The number of implanted atoms from the cluster is small, and, therefore, its influence is also small. All the clusters launched had six atoms and an octahedral (without an atom in the center) shape. The shape of the cluster does not determine the results, as the impacts on the surface were random, so they always occurred in a different way [14]. The potential used in the clusters is of the Lennard-Jones type (\(V\big{(}r_{ij}\big{)}=4\epsilon[(\sigma/r_{ij})^{m}-(\sigma/r_{ij})^{n}]\)) with \(n=6\) and \(m=12\), with suitable choices for the interatomic distance, \(\sigma\), and the depth, \(\epsilon\) (see the values in [15; 16] for He). In the case of He, which is a very light ion, the contribution of electronic stopping has been taken into account by means of another viscous force proportional to speed [17]. The surface is free of stress. A periodic FCC network at 5 K is heated to 300 K, scaling the temperature in steps. With each increase in temperature, the lattice relaxes in the canonical ensemble using an inertial factor \(Q=10^{-36}\) g cm\({}^{2}\)[18]. The final stress is removed by relaxing the system to zero pressure in the canonical isobaric ensemble with an associated inertia factor \(W=10^{19}\) g [19].