## Appendix A
The Coulomb repulsion in the electron gas affects both the rate of the electron avalanche and the dynamics of bubston expansion during the SOC process. It is clear that, at the stage of inertial retention (see below), avalanche development is governed by Equations (10) and (11) only for small average electron density inside the bubble \(\langle n_{e}\rangle=N_{e}/(4\pi{R_{0}}^{3}/3)\), i.e., when the free path length associated with electron-electron collisions is larger than \(R_{0}\). When the average density \(\langle n_{e}\rangle\) increases, the frequency of electron collisions with the wall must drop (\"peripheral\" electrons shield the wall from the collisions with \"central\" electrons) and consequently avalanche development ceases (electron-electron collisions do not contribute to the absorption of light). It follows from here that there should exist a maximum value of the average electron density, which should depend on the bubston radius, i.e., \(\langle n_{\rm{cmax}}\rangle=\langle n_{\rm{cmax}}\rangle\) (\(R_{0}\)). This value is to be determined. Electron gas inside a bubston must have non-uniform distribution owing to the Coulomb interaction, i.e., its density \(n_{e}=n_{e}(r)\), where \(r\) is the distance from the bubston center. The problem of determining \(n_{e}(r)\), and the corresponding electrical potential \(\varphi(r)\) distribution is the classical Debye problem, which can be solved under the condition \(n_{e}<<(T_{e}/e^{2})^{3}=2.7\times 10^{21}\) cm\({}^{-3}\) (see, for instance, [50]). Assuming the validity of the Boltzmann distribution for the electron density and considering this distribution,  
\[n_{e}(r)\ =\ n_{e}(0)\exp(3e\varphi/2T_{e})\approx n_{e}(0)(1+3e\varphi/2T_{e})\]  
in the area \(0<r<R_{0}\), we find that the Poisson equation \(\
abla^{2}\varphi=-4\pi\varphi=4\pi en_{e}(r)\) can be rewritten as  
\[\
abla^{2}\varphi-\varphi/a_{e}^{2}\ =\ 4\pi en_{e}(0), \tag{10}\]  
where  
\[a_{e}\ =\ \sqrt{T_{e}/6\pi e^{2}n_{e}(0)}\]  
is the Debye radius for electrons. Solving this equation, we should impose the following boundary conditions: the first condition \(\varphi(0)=0\), and the second condition  
\[4\pi\int\limits_{0}^{R_{0}}r^{2}n_{e}(r)dr\ =\ N_{e}. \tag{11}\]  
It is easy to show that the solution \(\varphi(r)\) satisfying the first condition is the following  
\[\varphi(r)\ =\ \frac{T_{e}}{e}\left(\frac{\sinh r/a_{e}}{r/a_{e}}-1\right),(0\leq r\leq R_{0}). \tag{12}\]  
For the distribution \(n_{e}(r)\) we obtain  
\[n_{e}(r)\ =\ n_{e}(0)\frac{\sinh r/a_{e}}{r/a_{e}}. \tag{13}\]  
In accordance with Equation (11), \(N_{e}\) has the following form  
\[dN_{e}/dt\ =\ N_{e}/\theta(I), \tag{10}\]0  
where \(x=R_{0}/a_{e}\). For the large values of \(N_{e}\), i.e., when \(x>1\), this equation can be presented as  
\[dN_{e}/dt\ =\ N_{e}/\theta(I), \tag{10}\]1  
According to Equations (10), (13) and (14), \(n_{e}(0)\) and \(n_{e}(R_{0})\)are equal to  
\[dN_{e}/dt\ =\ N_{e}/\theta(I), \tag{10}\]2  
It is clear that \(n_{e}(0)\) depends on \(N_{e}\) only logarithmically, and \(n_{e}(R_{0})\)= \(\langle n_{e}\rangle\ \times\ (x/3)\), where \(\langle n_{e}\rangle=N_{e}/(4\pi R_{0}^{3}/3)\) is the mean density of electrons. While \(N_{e}\) grows, electron gas is being compressed in the spherical layer of thickness \(a_{e}=R_{0}/x\) close to the bubble surface. In the case of a large value of \(N_{e}\), electrons are \"stuck\" in this layer and stop colliding with the wall. This happens when the electron free-path length in this layer \(l_{ee}\)= \(1/(\sigma_{ee}n_{e}(R_{0}))\) (here \(\sigma_{ee}\ =\ (9/16)\pi e^{4}/T_{e}^{2}\), see [51]) is equal to  
\[dN_{e}/dt\ =\ N_{e}/\theta(I), \tag{10}\]3  
and becomes less than the layer thickness \(a_{e}=R_{0}/x\), so we obtain for the mean electron density  
\[dN_{e}/dt\ =\ N_{e}/\theta(I), \tag{10}\]4This value is to be considered as the maximum (limiting) quantity, which restricts the electron avalanche development in the nanobubble.