# Multi-Directional Displacement Threshold Energy and Crystal Irradiation Damage Model
## 4 Atomic Displacement Model under Radiation  
### Development of Classical Models  
[PERSON] and [PERSON] introduced the concept of displacements-per-atom, which elucidates that those energetic particles predominantly travel in a straight path within a material but occasionally engage in strong binary collisions, transferring energy to lattice atoms. Consequently, they further postulated graft collisions as a sequence of binary elastic interactions between atoms and proposed a methodology for estimating the number of displaced atoms (\(N_{\text{d}}\)) generated by a PKA with energy \(E\). \[N_{\text{d}}= \begin{cases}0&0<E<E_{\text{d}}\\ 1&E_{\text{d}}<E<2E_{\text{d}}\\ E/2E_{\text{d}}&2E_{\text{d}}<E<E_{1}\\ E_{1}/2E_{\text{d}}&E_{1}<E<\infty\end{cases}  \tag{3}\]  
At energies above \(E_{1}\), the recoil only loses energy through electron excitation, while at energies below \(E_{1}\), the atoms are completely decelerated via hard-core elastic scattering. An atom that receives energy greater than the displacement threshold \(E_{\text{d}}\) is permanently displaced whereas an impacted atom receiving energy less than \(E_{\text{d}}\) eventually returns to its lattice position. The lattice effect is not considered in the process of single-atom displacement and cascade development. The Kinchin-Pease model assumes an irregular  
Figure 2: Schematic diagram of lattice damage by irradiation. (**a**) Complete lattice. (**b**) Close Frenkel pairs. (**c**) Distant Frenkel pairs. (**d**) Cascading-collision Frenkel pairs. arrangement of atoms in a solid, disregarding the influence of crystal structure and defining the probability of atomic displacement as a step function. \[P_{\text{d}}(T)= \begin{cases}0&T<E_{\text{d}}\\ 1&T>E_{\text{d}}\end{cases}  \tag{4}\]  
When utilizing this model, the fixed value is constrained within the range of 25 eV to 50 eV, with a preference for the lower value, which is commonly employed. This approximation proves highly advantageous when dealing with particles possessing minute cross-sections such as neutrons and electrons. In terms of ion and atom recoil, it is plausible for multiple simultaneous collisions to occur between neighboring atoms, resulting in what can be described as either a displacement spike or a heat spike. Building upon the Kinchin-Pease model, [PERSON] proposed a semi-empirical revision that incorporates several crucial modifications deemed significant [25]. \[N_{\text{d}}=\frac{\alpha\beta(E)W(E)}{\gamma(E)E_{t}} \tag{5}\]  
Here, \(\alpha\) is a factor introduced to incorporate realistic atomic scattering instead of the hard-core approximation, and it is generally recommended to maintain a value around 0.75. The factor \(\beta(E)\) was initially included to account for defect recombination within the cascade although it has often been disregarded by setting \(\beta(E)\) as equal to 1. The factor \(W(E)\) represents the fraction of initial PKA energy dissipated in elastic collisions. Focusing energy \(E_{f}\) serves as a parameter rather than displacement threshold \(E_{\text{d}}\). [PERSON] et al. improved the K-P model and derived the NRT model [10] as below:  
\[N_{F-P}= \begin{cases}0&0<E<E_{\text{d}}\\ 1&E_{\text{d}}<E<2E_{\text{d}}/0.8\\ 0.8E/2E_{\text{d}}&2E_{\text{d}}/0.8<E\end{cases}  \tag{6}\]  
The difference between the formula and the K-P model incorporates compound correction when calculating the number of defects generated by high-energy PKA. It is noteworthy that both the K-P model and the NRT model demonstrate a proportional relationship between the number of defects produced and PKA energy. In cases of higher-energy PKAs, a significant portion of energy is dissipated through atom displacement and excitation, with this proportion increasing as PKA energy rises. Conversely, due to loss cascades caused by high-energy PKAs, there exists a considerably high defect density within cascade regions. Within dozens of picoseconds, most defects undergo compounding off, leaving only surviving defects contributing to irradiation losses.