Streamer theory of breakdown in gases

The streamer theory (Raether and Meek, also Loeb) explains spark breakdown of gases at high $pd$ (above about 1000 torr·cm, e.g. long gaps in atmospheric air), where Townsend's theory fails. It states that a single electron avalanche, when it becomes large enough, distorts the field by its own space charge, and photo-ionization in the gas then produces a conducting plasma channel (streamer) that rapidly bridges the gap.

Why a new theory was needed

• Observed formative time lags ($\sim10^{-8}$ s) are much shorter than Townsend's prediction (time for ions to cross the gap, $\sim10^{-5}$ s).
• Breakdown at high $pd$ does not depend on the cathode material.
• Sparks are thin, bright, branched channels, not diffuse glows.

Mechanism

 (a) avalanche          (b) space charge
 C|  e- ->  ->  ->|A    C|      - - +++++|A
                         field enhanced near head

 (c) photo-ionization   (d) streamer
 C|   . <- . <- +++|A   C|<======plasma===|A
     secondary avalanches   channel bridges gap

1. Primary avalanche: a free electron near the cathode starts an avalanche ($e^{\alpha x}$ electrons) that moves to the anode.
2. Space charge: electrons move about 100 times faster than ions, so the avalanche head carries electrons while a cone of slow positive ions is left behind. When the avalanche reaches the anode, the electrons are absorbed and the positive space charge remains.
3. Field distortion: the space-charge field $E_r$ adds to the applied field near the space charge and becomes comparable to it once the avalanche has about $10^8$ charges.
4. Photo-ionization: excited atoms in the avalanche emit photons, which ionize gas molecules near the positive space charge, creating new electrons.
5. Secondary avalanches: these electrons start avalanches in the enhanced field and flow into the positive space charge, forming a quasi-neutral plasma. The positive tip moves forward.
6. Streamer propagation: the process repeats, so a conducting cathode-directed streamer grows very fast (about $10^8$ cm/s) towards the cathode. When it reaches the cathode, a highly conducting channel connects the electrodes: spark breakdown.

If the voltage is well above minimum, the avalanche can become critical mid-gap, and mid-gap streamers grow towards both electrodes.

Breakdown criteria

Meek's criterion: breakdown occurs when the radial space-charge field at the avalanche head equals the applied field ($E_r \approx E$):

$$E_r = 5.27\times10^{-7}\,\frac{\alpha\,e^{\alpha x}}{(x/p)^{1/2}}\ \text{V/cm}$$

Putting $E_r = E$ and $x = d$ gives the minimum breakdown voltage for the gap.

Raether's criterion: a critical avalanche size is needed:

$$\alpha x_c = 17.7 + \ln x_c \quad (\text{about } e^{\alpha x_c} \approx 10^8)$$

Features explained

The two theories complement each other: Townsend's mechanism applies to short, low-pressure gaps (low $pd$), and the streamer mechanism to long gaps at atmospheric and higher pressures.