## 3 Model
### Strategy for Obtaining Better Smoothness of Cut Walls
#### 3.2.7 Radius of Shock-Wave Modification  
In the mechanism of the modification zone formation by a shock wave, the plasma of each filament creates a so-called thermobaric effect: due to the development of the indicated above a high temperature of several tens of thousands of degrees, a high sub-megabar (or even megabar) pressure is achieved in the filament. Therefore, each such filament acts as a source of a cylindrical micro-explosion inside the material. The radial propagation of the shock wave changes the structure of the surrounding material, inducing its destruction. A certain modification zone is formed around the filament in the form of a cylinder with a radius \(R_{sw}\), about the same length \(H\), as that one of the filaments itself, and with a corresponding volume \(V=\pi R_{sw}{}^{2}H\). At the same time, we put also that in whole this modified zone, a threshold of mechanical stresses is reached not lower than a mechanical strength \(\sigma_{c}\). Thus, there is a destruction of the material and its corresponding mechanical weakening, sufficient to provide for a mechanical cleaving in the material after irradiation along the path of the laser beam movement. Let \(\gamma_{sw}Q_{sw}\) be the energy expended on the destruction of the material in the mentioned volume \(V\). Then, from the known approximate relation \(\sigma_{c}V=\gamma_{sw}Q_{sw}\)[27] and using the relation (40) for \(Q_{sw}\), the radius of critical modification by shock wave \(R_{SW}\) can be found as:  
\[R_{SW}=\left[\gamma_{sw}K_{sw}A_{P-H}\frac{(w-w_{dis})}{w_{mod}^{sw}}\right]^ {1/2}r_{f},\,\,\text{where}\,\,w_{mod}^{sw}=\sigma_{c}. \tag{46}\]  
This expression is functionally the same as the general expression (23) introduced above, if we put in (23) \(\gamma=\gamma_{sw}\), \(K=K_{sw}\), and \(w_{mod}=\sigma_{c}\). The conversion coefficient \(A_{P-H}\) is taken as previously in expressions (43-45) as \(A_{P-H}\approx 0.9\). Irreversible energy expenditures on the dissociation \(w_{dis}\) are also expected the same as previously. There is no precise information on the magnitude of the compressive strength \(\sigma_{c}\), for example, for chemically tempered glass Corning Gorilla Glass 5, only an estimate is known: \(\sigma_{c}\geq 0.85\) GPa [55], which in kJ/cm\({}^{3}\) is \(\sigma_{c}\geq 0.85\) kJ/cm\({}^{3}\). For greater certainty, we take into account, that, as is well known from the literature, for tempered glasses \(\sigma_{c}\) is 3-4 times higher than for non-tempered glasses, for which, in turn, the value of \(\sigma_{c}\) is known and according to reference data is about 0.59-0.6 GPa. Therefore, it can be assumed that for tempered glasses \(\sigma_{c}\) is about 1.8-2.4 GPa, i.e., 1.8-2.4 kJ/cm\({}^{3}\), and for sapphire \(\sigma_{c}\) about 2.4-3 GPa (=2.4-3 kJ/cm\({}^{3}\)). These values are given in Table 2.  
For crystalline materials, the threshold parameter \(\sigma_{c}\) in (46) may depend on the orientation of the filament axis relative to the crystallographic axes of the material. In particular, not the compressive strength, but the tensile strength, which is known to be 1-2 orders of magnitude lower than the indicated compressive strength, may be important. In this case, the radius \(R_{SW}\) can increase several times according to (46).  
Equation (46) assumes that the shock wave is initiated by the de-excitation for a sufficiently short time \(t_{E}\), i.e.:  
\[t_{E}<t_{P1}. \tag{47}\]In numbers according to the \(t_{P1}\) estimates in Section 3.1, this means that for the filament radius \(r_{f}\) =0.25-0.5 \(\mu\)m \(t_{E}\) < 80-170 ps. However, if the de-excitation is much slower:  
\[t_{E}>>t_{P2}, \tag{48}\]  
that is, according to the \(t_{P2}\) estimates in Section 3.1 for \(r_{f}\) = 0.25-0.5 \(\mu\)m, at \(t_{E}>>\) 0.4-0.8 ns, a significant decrease in the shock wave amplitude and a corresponding decrease in \(R_{SW}\) should be expected. \(R_{SW}\) estimates in this case are not included in this model.