## 3 Model
### Strategy for Obtaining Better Smoothness of Cut Walls
#### 3.2.1 Laser Pulse Energy  
To make the LFC process efficient and to impart to it a driving force, it is necessary to eliminate significant losses of filament energy to thermal radiation, which can spread far beyond the limits of the impact zone necessary for separation and thus do not contribute to its formation and propagation around the filament. To implement this factor, the VED \(w\) of the filament should be such that [39]:  
\[w\leq w_{1}\text{ or }w\geq w_{2}. \tag{12}\]  
In obtaining this result, we assume that the condition (10) is fulfilled. It means in this case that the filament temperature reaches its maximum vs time at about \(t\approx t_{E}\), when laser excitation of electrons, both in the free-electron plasma and in the electron-hole plasma, relaxes to a state equilibrated with temperature. Also, as according to (10) \(t_{E}\) is smaller of the heat confinement time \(t_{T1}\) in the filament, a practically all released heat appears inside the initial filament radius \(r_{f}\). Numerically, as follows from the given above estimates of \(t_{T1}\) for the submicron radius of the filament \(r_{f}\) = 0.25-0.5 \(\upmu\)m, this corresponds, for example, to \(t_{E}\) < 30-125 ns for silica glass and \(t_{E}\) < 6-40 ns for a sapphire. The VED bounds \(w_{1}\) and \(w_{2}\) in (12) correspond to some limits \(T_{1}\) and \(T_{2}\) for filament plasma temperature, \(T\leq T_{1}\) or \(T\geq T_{2}\), and:  
\[w_{1}\approx(1/K_{tw})C\rho(T_{1}-T_{i})+w_{dis},w_{2}\approx(1/K_{tw})C\rho( T_{2}-T_{i})+w_{dis}. \tag{13}\]  
Here \(C\) = \(C_{HT}\), and \(C_{HT}\) is a high-temperature (for \(T\) > 2500 K) specific heat. It is evaluated as \(C_{HT}\) = 3\(RN_{at}/\mu\)[50; 51], where \(R\) = 8.31 J/(mol\(\cdot\)K) is the gas constant, \(N_{at}\) is the number of atoms in the molecule, and \(\mu\) [g/mol] is a molar mass. For example, \(\mu\) = 60 g/mol and \(N_{at}\) = 3 correspond to both SiO\({}_{2}\) and SiO\({}_{2}\)-based glasses, and then one gets \(C_{HT}\) = 1.25 J/(g\(\cdot\)K). For sapphire (Al\({}_{2}\)O\({}_{3}\)) with \(\mu\) = 102 g/mol and \(N_{at}\) = 5 one obtains \(C_{HT}\) = 1.22 J/(g\(\cdot\)K). \(\rho\) is material density. \(w_{dis}\) is specific energy for material dissociation. \(T_{i}\) is initial (room) temperature. The dimensionless coefficient \(K_{tw}\) describes a part of the filament energy \((\pi r_{f}^{2}H)(w-w_{dis})\), which actually goes for material modification through the heat-wave channel (TW is the abbreviation for \"thermal wave\"). The other part of this energy, \(K_{sw}\), goes for the material modification through a shock-wave channel (SW is the abbreviation for \"shock wave\"). Based on the law of energy conservation we put:  
\[K_{tw}+K_{sw}=1. \tag{14}\]  
As \(T_{1}\), \(T_{2}\) >> \(T_{i}\), they are calculated as:  
\[T_{1}\approx[16(\text{In}2)^{1/2}\ \beta_{max}C\rho\chi/(\sigma r_{f})]^{1/3},\ T_{2}=2 Db/\lambda_{ ln}. \tag{15}\]  
The lower temperature limit \(T_{1}\), corresponding to \"not high\" plasma temperatures \(T\leq T_{1}\), is defined from the limiting allowed value of the ratio \(\beta\) of contributions of energy flux by thermal radiation (considered as \"useless\" losses) to the energy flux by heat conduction (considered as \"useful\" losses for formation and propagation of critical for separation modification zone). These fluxes are generated from the filament plasma during its cooling down. \(\beta_{max}\) denotes this limiting allowed value of the ratio \(\beta\). \(\chi\) is material thermal diffusivity. On the other hand, \(T_{2}\) is a \"high enough\" temperature, at which and above which the main part of filament energy, although emitted as thermal radiation, but is not lost from the affected zone, as at so high temperature this wide-spectrum thermal radiation appears for more than 90% of its energy to be inside the spectrum of \"strong\" absorption (\(\alpha\geq 10^{5}\ \times\ \text{cm}^{-1}\)) of the surrounding cold material.