# Multi-Directional Displacement Threshold Energy and Crystal Irradiation Damage Model
## 4 Atomic Displacement Model under Radiation
### New Method for Displacement Probability
At every point along its trajectory representing its motion path upon impact, we superimpose interactions between this moving atom and all its nearest neighbor atoms, summing up interaction energies from the three adjacent face centers connected by the imaginary lines on the graph. At the center of the triangle formed by the dotted lines, total potential energy reaches its maximum value: namely, our barrier saddle point, with an energy level denoted as \(\epsilon^{*}\) (as shown in Figure 4a). The difference between \(\epsilon^{*}\), representing energy at the saddle point, and \(\epsilon_{\text{eq}}\), representing the equilibrium position within the potential well, signifies displacement threshold energy within this particular direction. The displacement threshold energy of Ga and As atoms along the six highly symmetrical directions [001], [110], [111], [103], [123], and [112] was calculat  
Figure 4: Out-of-position threshold energy and orientation in a crystal structure. (**a**) [111] direction displacement threshold energy obtained from the difference between the energy at the saddle point and the energy of the potential well at the atomic equilibrium position. (**b**) High-symmetry direction in the GaAs bulk phase [26]. Figure 3: The situation of recoil-atom displacement in sphalerite structures. the simulation, one of these six directions was randomly selected as a constraint condition for each collision. The high-symmetry direction in the GaAs bulk phase is illustrated in Figure 4b, and the corresponding results are presented in Table 1.  
Therefore, in the sphalerite structure, the energy received (\(E\)) by the impacted atom during a collision is considered as the radius of a circle. This circle intersects with the Sine Surface to generate both stable and displacement regions, as depicted in Figure 5 The ratio of the displacement region to the entire sphere represents the probability of displacement for the impacted atom. The probability distribution function of atomic displacement is established as below:  
\[P_{\text{d}}(T)= \begin{cases}0&T<E_{\text{d,min}}\\ \frac{\omega_{\text{c}}}{\omega_{\text{E}}}&E_{\text{d,min}}<T<E_{\text{d,max}} \\ 1&T>E_{\text{d,max}}\end{cases}  \tag{10}\]  
The energy \(T\) carried by the atom is considered to be the radius of a sphere due to the random collision angle. As the number of collisions increases, the radius gradually decreases. When it falls below the minimum displacement threshold energy \(E_{\text{d,min}}\), no displacement occurs. Conversely, when it exceeds the maximum threshold energy \(E_{\text{d,min}}\), displacement becomes inevitable. For recoil atom energies within these upper and lower bounds, their probability can be determined by calculating the ratio of the spherical area \(\omega_{\text{c}}\) intersected with a sine surface to \(\omega_{\text{E}}\), which represents the entire sphere. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Displacement Threshold in GaAs (eV)**} \\ \hline Crystal Orientation & Ga & As \\
[001] & 14.5 [26], 14 [27] & 10 [26], 16 [27] \\
[110] & 12 [26], 16 [27] & 10 [26], 20 [27] \\
[111] & 12 [26], 16 [27] & 8.5 [26], 16 [27] \\
[013] & 17.5 [26] & 12 [26] \\
[112] & 22.5 [26] & 10 [26] \\
[123] & 30 [26] & 10 [26] \\ \hline Lattice constant & \multicolumn{2}{c}{5.66 [28], 5.65 [29]} \\ Cohesive energy & \multicolumn{2}{c}{3.7 [30], 3.18 [31], 3.35 [29]} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Displacement threshold energy, lattice constant, and cohesive energy of GaAs. Figure 5: Displacement probability model of impacted atom. The specific expression of Sine Surface function as the threshold limit of atomic collision probability model in GaAs is given below:  
\[\left\{ \begin{array}{l}x=a\sin u\\ y=a\sin v\\ z=a\sin(u+v)\\ a=a_{0}\sqrt{E_{\rm{coh}}^{2}+E_{\rm{dis}}^{2}}\end{array} \right. \tag{11}\]  
Here, \(a_{0}\) is the lattice constant and \(E_{\rm{dis}}\) is the displacement threshold energy in different directions.