# Multi-Directional Displacement Threshold Energy and Crystal Irradiation Damage Model
## 4 Atomic Displacement Model under Radiation
### New Method for Displacement Probability  
The atoms occupying the regular lattice positions are situated within stable potential wells. In order for the colliding atoms to escape their respective potential wells, they must overcome the potential barrier created by the surrounding atoms. The highest point of this barrier is referred to as the saddle point while the difference in height between the saddle point and the lowest point of the potential well represents the barrier height or departure threshold energy. Due to its inherent directional and symmetrical nature, a crystal structure exhibits varying atomic barrier heights in all directions around its equilibrium position. Assuming that any further collisions by a primary knock-on atom (PKA) have no effect at energies below displacement threshold, Equation (8) describes the energy after \(n\)th collision. \[T_{n}=\left(1-\frac{4M_{1}M_{2}}{(M_{1}+M_{2})^{2}}\sin^{2}\frac{\alpha}{2} \right)\times T_{n-1} \tag{8}\]  
Here, \(M_{1}\) and \(M_{2}\) are mass of particles in collision. And, transferred energy \(T\) reaches maximum at scattering angle \(\alpha=0\). The collision process is regarded as terminated when \(T_{n}\leq E_{\text{d}}\). It is evident that the variability in recoil angle \(\alpha\) leads to variations in the number of displacement atoms, even when subjected to the same incident energy. The extent of damage inflicted on the crystalline medium is directly proportional to the uniform distribution of this energy. In Monte Carlo simulation, the number of collisions generated by each atom from its initial collision with other atoms until its energy falls below an offsite threshold determines the number of iterations since \(T_{n}\leq E_{\text{d}}\). When an atom is displaced, a new iteration with its energy denoted as \(T_{1}\) and the count is re-evaluated. The total number of displaced atoms can be obtained by summing the product of the number of impacted atoms at each instance and their corresponding displacement probability (\(P_{i}\)). \[N_{\text{d}}=\sum_{i=1}^{n}N_{i}\cdot P_{i} \tag{9}\]  
The crystal structure of GaAs consists of two phases, namely sphalerite and monoclinic structures, with the latter being obtainable only under high-pressure conditions. The sphalerite structure belongs to the cubic crystal system, where cations are positioned at the center node along the vertical axis, resulting in a coordination number of 4 for both anions and cations. By considering the face-centered cube as a body-centered cube with stronger constraints on nearest-neighbor distance differences, it is observed that the probability of deviating from its original position is significantly lower than that of moving towards void directions among the eight connecting bond directions. The atoms in the normal lattice position are confined within a stable potential well. For the energetically excited atoms to escape this potential well, they must overcome the barrier created by their surrounding atoms. The highest point of this barrier is known as the saddle point. The difference in height between the saddle point and the lowest point of the potential well represents the barrier height, which serves as an energy threshold for out-of-position events. Due to specific crystallographic directions and symmetries inherent in crystal structures, atomic barrier heights exhibit slight variations across different directions around equilibrium positions. If we possess knowledge of the interaction potential function between lattice atoms, it is theoretically feasible to calculate each crystallographic direction's respective barrier height or off-site threshold energy value. Let us assume that after a collision event, an atom located at the lower-left corner of Figure 3 gains energy and moves along the [111] direction.