# Multi-Directional Displacement Threshold Energy and Crystal Irradiation Damage Model
## 2 Displacement Threshold  
The phenomenon of atomic displacement occurs when a lattice atom in its normal position is struck and acquires energy exceeding a certain threshold, causing it to depart from its initial lattice site. Such an atom is referred to as an off-site atom. However, not all atoms can undergo delocalization upon receiving any amount of energy; only when the energy imparted to the impacted atom equals or exceeds a specific minimum value can it become delocalized. This minimum energy required for atomic displacement is defined as the threshold energy for atomic displacement. When radiation particles collide with solid materials and lattice atoms, a series of collision processes ensue, which can be categorized into two distinct stages. The primary process involves direct interaction between incident particles and lattice atoms in the solid, resulting in the transfer of energy to the struck atoms. These directly impacted atoms, recoiled from their original positions, are referred to as primary knock-on atoms (PKA). Primary knock-on atoms with sufficient energy to exit their lattice positions are known as primary off-site atoms. The secondary process is that when the incident particle decelerates inside the solid, multiple primary collision atoms with different energies can be generated. Because these primary collision atoms have considerable kinetic energy, they impact other lattice atoms and delocalize them to form secondary collision atoms. A PKA refers to an atom that is directly ejected from a material due to energy transfer during interaction with radiation. If the energy is high enough, more than one atom can be ejected (although these are not considered PKAs but, rather, damage cascades). The generation of PKAs depends on various factors including ray type, ray energy, and other considerations. Different types of rays such as neutrons, ions, and electrons exhibit distinct behaviors in materials, requiring separate treatment methods for their interactions. When an atom leaves its lattice position it must overcome a barrier much larger than its binding energy, known as \(E_{\mathrm{d}}\) (Displacement Energy). If the energy transferred to the atom exceeds the displacement threshold energy, it can potentially become a primary knock-on atom (PKA) [21]. In this case, the atom is displaced from its initial position and loses an amount of binding energy (\(E_{\mathrm{b}}\)) due to overcoming crystal lattice forces, resulting in a kinetic energy of \(E_{\mathrm{t}}\)-\(E_{\mathrm{b}}\). When the displacement distance of the atom caused by the PKA is greater than one lattice constant, it can induce further displacement damage (in cases with high PKA energy) or create a vacancy defect (in cases with low PKA energy). Conversely, if the displacement atom does not possess sufficient energy, it may be knocked out of its original position but subsequently pulled back by neighboring atoms due to insufficient escape velocity. The excess gained energy then contributes to lattice vibrations known as phonons without generating a PKA. After the generation of a primary knock-on atom (PKA), the process of radiation damage in materials involves the interaction between PKAs and material crystals, which remains consistent across different types of radiation. This observation underscores the universality of studying post-PKA generation damage processes wherein various types of radiation can be employed for investigating radiation-induced damage. The extremely short duration of PKA generation, typically ranging from a few femtoseconds to dozens of femtoseconds, renders it impractical for conducting experimental research using current techniques. Hence, computer modeling methods are widely employed. As PKAs are essentially partially ionized particles, their interactions with materials can be simulated using the Monte Carlo software (MCNP, Geant4, SPECTER, CASINO) or molecular dynamics techniques.