# 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  
For certain indicators, there exists a disparity in the measurement outcomes among researchers across different time periods; however, this discrepancy also provides a range of values. Apart from variations in experimental conditions, the alteration of bond energy due to atomic vibrations during collisions contributes as a significant factor. High-energy particles experience successive collisions as they pass through the medium. After each collision, atoms exhibit specific vibrational frequencies without undergoing displacement, resulting in alterations in the bond energy between neighboring atoms and the displacement threshold energy in various directions. As a result, the likelihood of subsequent displacements occurring at the same position is modified based solely on prior collisions. This phenomenon can be considered as the transfer of different states between two data, and these discrete alterations are regarded as Markov processes. The nth collision results in the representation of displacement threshold energies corresponding to each direction utilizing state vector \(S_{n}\). \[S_{n}=[S_{n,1}\ \ S_{n,2}\ \cdots\ S_{n,i}\ \ S_{n,i+1}] \tag{13}\]  
To transfer state \(S_{n}\) to state \(S_{n+m}\) within \(m\) collisions, a state transition matrix is needed, which can be described as follows:  
\[p= \begin{bmatrix}p_{11}&p_{12}&\ldots&p_{1n}\\ p_{21}&p_{22}&\ldots&p_{2n}\\ \ldots&\ldots&\ldots&\ldots\\ p_{n1}&p_{n2}&\ldots&p_{nn}\end{bmatrix}  \tag{14}\]  
\(P_{ij}\) denotes the probability of a single particle moving from state \(i\) to state \(j\) within a unit step. Since matrix \(P\) is a first-order isotropic Markov chain transfer matrix, it should satisfy the following condition:  
\[\sum_{j=1}^{n}p_{ij}=1,\,p_{ij}\geq 0 \tag{15}\]  
It is worth noting that, using the [111] direction as an illustrative example, the maximum and minimum values reported in the previous literature represent state limits along with the presence of a displacement-threshold-energy probability transfer matrix. The interval from 8.5 to 16 is uniformly divided into multiple segments (with smaller intervals corresponding to higher incident particle energies). During simulation, a random value is selected as the displacement threshold energy, resulting in a state transition after each collision. The transition probability is assigned equally to different states based on their distances, with states ranging from 8.5 eV to 16 eV being more likely to become adjacent states after collision. Figure 6 exhibits a bias towards the elastic barrier model wherein it tends to move towards the boundary states at both ends and either returns to an adjacent state or remains in the boundary state. The relationship between spacing and transition probability is described using a Gaussian distribution, indicating a preference for transitions towards adjacent states and higher likelihood of rebounding back to adjacent states upon reaching the boundaries. In cases of low-energy electron irradiation with minimal changes in displacement threshold energy after each impact, a circular random walk should be employed (refer to Figure 7). For materials with more metallic properties, or where the operating environment has a large temperature fluctuation (long period, low amplitude annealing process), atoms in lattice gaps can re-enter nearby vacancies, as shown in Figure 8. Initially, a significant number of atoms undergo displacement; however, upon the cooling down of the cascade, nearly all atoms reoccupy their ideal crystal positions due to the non-thermal recombination effect [32]. Figure 6: Elastic barrier random walk model state diagram and its probability transition matrix. Figure 7: Circular random walk state diagram and its probability transition matrix. Clearly, a significant proportion of atoms do not return to their initial positions after irradiation.