## 3 Model
### Strategy for Obtaining Better Smoothness of Cut Walls
#### 3.2.6 The Radius of the Modification Zone under the Action of Heatwave
Note that when using the above assumption about the Gaussian distribution of \(w\) in the filament, a slightly different description of the temperature is also possible, which, in contrast to (32), is more physically correct, since it is formally limited at \(t\to 0\)[30]:  
\[T(r,\ t)-T_{i}=\frac{Q}{\pi r_{g}^{2}(t)HC\rho}\exp\left(-\frac{r^{2}}{r_{g}^{ 2}(t)}\right),\ \ r_{g}(t)=\left(r_{f}^{2}+4\chi t\right)^{1/2}.\]  
Here \(r_{g}(t)\) is a Gaussian (1/e)-radius of the temperature profile around filament after pulse termination. As seen, \(r_{g}\) increases with time due to radial heat conduction. However, for the times defined by the condition (28) \(t>>t_{T1}=r_{f}^{2}/(4\chi)\) that are of interest to us according to the condition (2), we can neglect the term \(r_{f}^{2}\) in comparison with \(4\chi t\) (\(r_{f}^{2}<<4\chi t\)) in the expression for \(r_{g}(t)\). Then this solution for \(T\) practically coincides with the more convenient for performing calculations and simpler solution (32), which is therefore used further for the necessary calculations. Under conditions (27) and (33), we first consider a relatively simple case where the pulse repetition time \(t_{rp}\) is much greater than the cooling time of the shown in Figure 5a point \(r=R\). This means that the next pulse coming in the time \(t_{rp}\) and creating the next filament (Y\({}_{2}\) in Figure 5a), from which the specified point \(r=R\) is at the same distance \(R\) (since when condition (22) is fulfilled and \(s_{1}=s_{1 max}=2R\) this point is in the middle between the filaments Y\({}_{1}\) and Y\({}_{2}\)), induces heating in the point \(r=R\) that has already completely cooled down from the first pulse. Reaching the threshold value \(T=T_{mod}\) is then determined by the action of the heat release from only one filament. In the opposite case, when the pulse repetition time \(t_{rp}\) is sufficiently short compared the time for cooling down from the first pulse, the heating at the point \(r=R\) is formed by heat release from two adjacent filaments, as shown in Figure 5b, which will also be discussed below. The radius of the modification zone critical for separation under these conditions is defined as the maximum radial distance of penetration into the material of the isotherm \(T=T_{mod}\) from the filament axis. Verification of compliance with conditions (27) and (33) is possible if data on the value of \(t_{E}\) for this material is available only after performing calculations of \(R\) and \(s_{1}\). To calculate the radius of the modification zone around the filament we take the fixed \(r=r_{1}\) in the expression (32) for the temperature. Considering then \(T(r_{1},t)\) as a function of time only, from the condition \(dT(r_{1},t)/dt=0\), we can easily find the position in time \(t_{max}\) of the temperature maximum \(T=T_{max}\) at the radius \(r_{1}\):  
\[t_{max}=r_{1}^{2}/(4\chi). \tag{34}\]At \(t>t_{max}\), the cooling of the point \(r=r_{1}\) starts, that is, its temperature decreases. It follows from (34) that the relation \(r_{1}=2(\chi t_{max})^{1/2}\) is fulfilled. Therefore, reaching the maximum temperature at the radius \(r_{1}\) can be interpreted as an arrival to this point from the heated filament of thermal diffusion wave, which has a length of \(2(\chi t_{max})^{1/2}\). The substitution of \(r=r_{1}\) and \(t=t_{max}\) with the use of (34) into (32) gives the maximum temperature \(T_{max}(r_{1})\):  
\[T_{max}(r_{1})-T_{i}=\frac{1}{\rm e}\ \frac{Q}{\pi r_{1}^{2}HC\rho}. \tag{35}\]  
It is seen from here that the heating \([T_{max}(r_{1})-T_{i}]\) decreases \(\propto\)1/\(r_{1}\)\({}^{2}\) with increasing the distance \(r_{1}\).