# Multi-Directional Displacement Threshold Energy and Crystal Irradiation Damage Model
## 4 Atomic Displacement Model under Radiation
### The Displacement Threshold Energy Transfer Probability of Different Crystal Types
The number of atom substitutions far exceeds the number of defects generated, indicating a higher probability of defect formation compared to maintaining the original shape. Consequently, as irradiation progresses, the expected displacement threshold energy for colliding particles gradually decreases. In order to rectify this issue, it is necessary to transform the transition probability from being constant into a function. [PERSON] and [PERSON] propose that the evolution of the semi-Markov process {r(t), \(t\geq 0\)} is governed by the following probabilistic transitions [33]. \[\Pr\{r(t+h)=j\mid r(t)=i\}=\left\{ \begin{array}{l}\lambda_{ij}(h)h+o(h),\,r(t) \text{ jumps from }i\text{ to }j\\ 1+\lambda_{ii}(h)h+o(h),\text{ otherwise}\end{array} \right. \tag{16}\]  
Here, {r(t), \(t\geq 0\)} is a continuous-time semi-Markov process taking values in a finite space; \(S=\{1,\,2,\,\cdots,\,N\}\), \(\lambda_{ij}(h)\) is the transition rate from mode \(i\) to mode \(j\) at \(t\) when \(i\
eq j\), \(\lambda_{ii}(h)=-\sum\limits_{j=1,i\
eq j}^{N}\lambda_{ij}(h)\), and \(o(h)\) is notation defined such that \(\lim_{h\to 0}\frac{o(h)}{h}=0\). The GaAs solar cells in the space power system are subjected to high levels of proton irradiation energy, but their material does not undergo significant changes in electron trap and hole trap characteristics when annealed at low temperatures [34]. Therefore, the modified Elastic Barrier Random Walk (EBRW) model is suitable for describing the variation in crystal departure threshold energy. Figure 9 illustrates the number of displacement atoms obtained using different methods. Compared to the other three linear models, our proposed method aligns more closely with the arc-NRT model. Additionally, the gradual increase in proton irradiation at low energies reflects the uncertainty associated with multi-directional ionization threshold energy during collisions. Utilizing protons with different energies to produce primary knock-on atom (PKA) energies of 50 keV, 100 keV, and 150 keV as per Equation (12), the results depicted in Figure 10 illustrate that after 10,000 repeated impacts, the proposed method consistently presents a lower count of F-pairs in comparison to the predictions of the NRT model. This discrepancy can be attributed to incorporating displacement threshold energy and defect reset into our approach. Furthermore, the study revealed a higher concentration of atom distribution under various incident proton energies compared to more dispersed results obtained through SRIM calculations. Figure 8: Collision cascade induced by a 10 keV recoil in Fe at an ambient temperature of 0 K [32]. Figure 10: Distribution of defects and recoil atoms in GaAs irradiated by protons with different energies. **(a)** Defect distribution for 50 KeV PKA impact. **(b)** Defect distribution for 50 KeV PKA impact. **(c)** Probability distribution under the proposed method for 50 KeV PKA impact. **(d)** Defect distribution for 100 KeV PKA impact. **(e)** Defect distribution for 100 KeV PKA impact. **(f)** Probability distribution under the proposed method for 100 KeV PKA impact. **(g)** Defect distribution for 150 KeV PKA impact. **(h)** Defect distribution for 150 KeV PKA impact. **(i)** Probability distribution under the proposed method for 150 KeV PKA impact. Figure 9: Variation rule of defect number under different methods.