Electron avalanche

Equations 2.7 and 2.11 can be combined into a more general expression which can be solved to determine the change in electron density due to both $\nu_i$ and $\nu_a$:

$$\frac{dn_e}{dt} = (\nu_i-\nu_{a})n_e$$ (2.13)

$$n_{e}(t) = n_{o}e^{(\nu_i-\nu_{a})t}$$ (2.14)

It is clear from Equation 2.14 that the electron density will increase exponentially if $\nu_i\!>\!\nu_a$. This occurs when $E/N \gtrsim 123$ Td in Figure 2.1. An exponential increase of electrons from one or more seed electrons is an electron avalanche.

An electron avalanche will develop at the electron drift velocity, $\vec{v}_e$. As it develops, the avalanche will expand due to diffusion and mutual electrostatic repulsion. The diffusion expansion dominates during the initial stages while electrostatic repulsion dominates in the latter stages as a result of the growing number of electrons (Bazelyan and Raizer, 1997, p. 80). An electron avalanche will also leave behind a trail of relatively immobile positively charged ions. A schematic illustration of an electron avalanche is shown in Figure 2.2a.

Figure 2.2: a) A schematic diagram of an electron avalanche. The electrons leave behind a trail of positive ions as they propagate towards the positively charged anode. b) An actual photograph of individual electron avalanches. (From Raether (1964)).

In a laboratory experiment where $n_o$ is sufficiently small, electron avalanches starting from a single electron can be clearly distinguished on a photograph. Figure 2.2b shows a photograph from Raether (1964) of electron avalanches which began in this fashion. Electron avalanches can transition into streamers and are also a fundamental component of streamers, as will be discussed in Section 2.6. The relevance of electron avalanches and streamers to sprites will be shown in Chapter 5.