Neon Light Boom

Noble gases like Neon $Ne$ as a $T^{-}$ and $g^{+}$ collector in a Sun capture lead us to Townsend discharge and Paschen's law.

This could be $v_{boom}$ a phenomenon where a voltage $V$ between a gap of length $d$ span an electric force $E=\cfrac{V}{d}$ that accelerates free ions to $v_{boom}$ and generates basic particles.

Intuitively, a higher voltage is needed for small gap to accelerate the particles to $v_{boom}$ before covering the length $d$,

$v^2=2ax$

$x\le d$

from $v^2=u^2+2ax$,

where $a$ is the acceleration under the constant field $E=\cfrac{V}{d}$ and we assume that the particle has zero initial velocity.

$F=qE$

so,

$a=\cfrac{F}{m_i}=\cfrac{qE}{m_i}=\cfrac{qV}{m_id}$

thus,

$v^2=2ax=2\cfrac{qV}{m_i}\cfrac{x}{d}$


If $v=v_{boom}=3.4354*\cfrac{density}{Z}$  and

$PV_{ol}=nRT$

$P=\cfrac{n}{V_{ol}}RT=\cfrac{M}{m_iN_AV_{ol}}RT=\cfrac{density}{m_i}kT$

where $N_A$ is the number per mole, $M$ is total mass of the gas.  So

$v=v_{boom}=3.4354*\cfrac{m_iP}{ZkT}$

and

$\left(3.4354*\cfrac{m_i}{Zk}\right)^2\left(\cfrac{P}{T}\right)^2=2\cfrac{qV}{m_i}\cfrac{x}{d}$

$V\cfrac{x}{d}=\cfrac{3.4354^2}{2}\cfrac{m_i^3}{qZ^2k^2}\left(\cfrac{P}{T}\right)^2$

$x\le d$

$V=V_{min}$  when

$x=d$

$V_{min}=\cfrac{3.4354^2}{2}\cfrac{m_i^3}{qZ^2k^2}\left(\cfrac{P}{T}\right)^2$

which is simply quadratic,

$V_{min}=A\left(\cfrac{P}{T}\right)^2$

where $A=\cfrac{3.4354^2}{2}\cfrac{m_i^3}{qZ^2k^2}$

This assumes that after attaining $v_{boom}$, collisions that generates particles is a certainty before reaching the other electrode, which may not be true for very low pressure and small gap length.

Strictly speaking, a discharge occurs,

$V\gt V_{min}$

$V\gt A\left(\cfrac{P}{T}\right)^2$

where $A=\cfrac{3.4354^2}{2}\cfrac{m_i^3}{qZ^2k^2}$

Maybe...