Limitations of Townsend theory; Paschen's law

Limitations of Townsend's theory

Townsend's theory (avalanche growth by $\alpha$, sustained by cathode secondary emission $\gamma$, breakdown when $\gamma e^{\alpha d} = 1$) explains breakdown well at low $pd$, but fails in several respects at higher $pd$ (above about 1000 torr·cm, e.g. long gaps in atmospheric air):

1. Time lag: Townsend predicts formative times of the order of $10^{-5}$ s (time for ions to cross the gap many times). Observed times in long gaps are about $10^{-8}$–$10^{-7}$ s, far too short for ion transit.
2. Cathode independence: At high pressures breakdown voltage is nearly independent of cathode material, which should not happen if $\gamma$ at the cathode controls breakdown.
3. Channel shape: Townsend predicts a diffuse glow-type discharge, but actual sparks at high $pd$ are thin, bright, branched and zig-zag channels.
4. Space charge ignored: The theory neglects the distortion of the field by the space charge of the avalanche, which becomes significant when the avalanche contains about $10^8$ electrons.
5. Non-uniform fields: It does not explain corona and breakdown in non-uniform fields, or polarity effects.
6. Breakdown with no cathode involvement: Discharges can start mid-gap and grow towards both electrodes, which Townsend's mechanism cannot explain.

These limitations led to the streamer (Raether–Meek) theory for high $pd$.

Derivation of Paschen's law

Townsend's breakdown criterion:

$$\gamma\left(e^{\alpha d} - 1\right) = 1 \;\Rightarrow\; \alpha d = \ln\left(1 + \frac{1}{\gamma}\right) \qquad (1)$$

Experimentally, the first coefficient depends on $E/p$:

$$\frac{\alpha}{p} = A\, e^{-Bp/E} \qquad (2)$$

where $A$ and $B$ are gas constants. In a uniform field $E = V/d$, so at breakdown $E = V_b/d$:

$$\alpha = Ap\, e^{-Bpd/V_b} \qquad (3)$$

Substitute (3) in (1):

$$\begin{aligned} Apd\, e^{-Bpd/V_b} &= \ln\left(1 + \frac{1}{\gamma}\right) = K \\ e^{-Bpd/V_b} &= \frac{K}{Apd} \\ \frac{Bpd}{V_b} &= \ln\frac{Apd}{K} \\ V_b &= \frac{B\,pd}{\ln (A\,pd) - \ln K} \end{aligned}$$ $$\boxed{V_b = \frac{B\,(pd)}{\ln\left[\dfrac{A\,(pd)}{\ln(1 + 1/\gamma)}\right]} = f(pd)}$$

Since $A$, $B$ and $\gamma$ are constants of the gas (and electrode), the breakdown voltage of a uniform-field gap is a function of the product $pd$ only. This is Paschen's law. For air at temperature other than 20 °C, $p$ is replaced by gas density ($Nd$ or $\delta d$).

Minimum sparking voltage

Setting $\dfrac{dV_b}{d(pd)} = 0$:

$$(pd)_{min} = \frac{e}{A}\ln\left(1 + \frac{1}{\gamma}\right), \qquad V_{b,min} = \frac{eB}{A}\ln\left(1 + \frac{1}{\gamma}\right)$$

($e = 2.718$.) For air, $V_{b,min} \approx 327$ V at $pd \approx 0.57$ torr·cm.

 Vb
  |\                      /
  | \                  /
  |  \             /
  |   \_______ /
  |      min ~327 V (air)
  +---------------------------> pd

Left of the minimum, too few collisions; right of the minimum, too little energy per collision; in both cases $V_b$ rises.