# Multi-Directional Displacement Threshold Energy and Crystal Irradiation Damage Model
## 3 Crystal Defect Types and Formation  
In an ideally complete crystal, the atoms are arranged in a strictly ordered manner on regular and periodic lattice points in space. However, during the actual growth process, crystal defects can arise due to changes in temperature, pressure, and medium composition concentration. Additionally, some defects may be generated as a result of thermal motion or particle stress after crystal formation. These defects have the ability to migrate within the lattice and even disappear while new defects can also emerge. They represent irregularities or imperfections in the arrangement of particles within the crystal structure and are commonly referred to as lattice defects. It is evident that within localized regions of the crystal structure, particle arrangement deviates from the periodicity of repeated space lattices and appears disordered. When incident particles interact with solid materials, they strike lattice atoms, causing them to move from their initial positions continuously until they occupy interstitial or vacancy positions, forming various types of defect states [23]. Based on the distribution range of disordered arrangements, defect types can be categorized into body defects, surface defects, line defects, and point defects [24]. When the temperature deviates from zero, atoms invariably exhibit thermal vibrations and acquire energy from this motion. Despite the substantial barrier that must be overcome for an atom to depart from its lattice position, defects can still arise due to the tunneling effect. Consequently, in a perfect crystal (devoid of impurities), a dynamic equilibrium is achieved between defect generation and recombination as the temperature exceeds 0 K. This thermally driven balance can be estimated using statistical methods to determine defect concentrations. The energy acquired by atoms through lattice vibrations follows the Boltzmann distribution. \[N_{\text{E}}=A\exp\left(\frac{-E}{kT}\right) \tag{1}\]  
Here, \(E\) denotes the kinetic energy associated with thermal motion, \(N_{\text{E}}\) represents the count of atoms possessing energy \(E\) at a given temperature, \(k\) is the Boltzmann constant (\(k\approx 1.38\times 10^{-23}\) J/K), \(T\) signifies the absolute temperature, and \(A\) stands for a constant. Figure 1: The displacement energy varies with the displacing direction in the carbon nanotube [22]. The determination of defect population in an equilibrium perfect crystal can be inferred from the Boltzmann distribution, employing vacancy as an illustrative example. \[N_{\rm v}=N\exp\left(\frac{-Q_{\rm v}}{kT}\right) \tag{2}\]  
In this formula, \(N\) represents the total number of atoms in the crystal lattice while \(N_{\rm v}\) denotes the number of vacancies present. \(Q_{\rm v}\) signifies the formation energy associated with vacancy creation, which is a material-specific property typically ranging from 0.3 eV to 3 eV. When the energy \(E\) received by the atom is below a certain threshold, it becomes confined to its initial lattice position due to hindrance from surrounding atoms and eventually dissipates through irregular thermal vibrations. In such cases, point defects cannot be formed. When the energy slightly exceeds this threshold, it becomes possible for the atom to overcome atomic hindrance and create a vacancy at its original lattice site. However, due to limited energy carried, significant displacement from the initial position is not feasible; instead, it remains in close proximity to the vacancy, forming a nearby Frenkel defect pair (Figure 2b). For higher values of energy \(E\), substantial distances can be traversed within the solid before settling into a specific gap position, resulting in long-distance Frenkel defect pairs (Figure 2c). If an atom acquires sufficiently high energy \(E\) upon impact, it not only displaces itself but also triggers the dislocation of other lattice atoms, leading to a cascade of collisions. Eventually, after all moving atoms come to rest, several Frenkel defect pairs are formed (Figure 2d).