paschen law
- 2. 1 .Objective : If a voltage differential is supplied to a gas as shown in the setup (Figure 1), an electric field is formed. If the electric field applied is strong enough, an avalanche process (the Townsend avalanche) is started which leads to the breakdown of the gas and the formation of plasma. This document describes the physics of this process and derives the law (Paschen’s law) that predicts the voltage differential that needs to be supplied in order to create the plasma.
- 3. 2. Description: As the voltage difference is applied, the electric field will accelerate any free charges that exist. Free electrons exist in the system due to random events from a variety of mechanisms including the triboelectric effect or through astronomical particles traversing the vessel and ionizing neutral particles. If an electron can gain more than the ionizing energy of the gas, UI (approximately 14eV for Nitrogen), the electron can ionize the neutral particle and create a new free electron and a free ion. The new free electron can repeat the process and create a chain reaction called the Townsend Avalanche. The ion is accelerated towards the cathode and as it hits it there is a chance that it will
- 5. 4 Derivation of Paschen’s Law: Quantitatively, the electron avalanche can be described using the rate of ionization per unit length, α. For example, if α = 2cm−1 then within a cm of electron travel along the x- axis, there will, on average, be 2 collisions with neutral particles that result in ionizing events. This results in the following equation describing the increase of electron current density, Γe(x): dΓe(x) = Γe(x)αdx, ……(1)
- 6. which integrates to: Γe(x) = Γe(0)e αx …….(2) Since there is no current leaving or entering the vessel except through the electrodes, and there is no charge accumulation, the continuity equation results in: Γ(0) = Γ(d),……………. (3) that is, the current density at the cathode (x = 0) is the same as that at the anode (x = d). Assuming a single ion species, Equation 3 can be rewritten as: Γe(0) + Γi(0) = Γe(d) + Γi(d)……. (4) Γi(0) = Γe(d) − Γe(0) + Γi(d)……(5)
- 7. Using the fact that there is no input of ions from the anode, Γi(d) = 0. Using this constraint, and substituting Equation 2 into Equation 5 results in: Γi(0) = Γe(0)(e^ αd − 1)……(6) As an ion reaches the cathode, the probability of it releasing a secondary electron is given by the coefficient γ, therefore: Γe(0) = γΓi(0)…...(7)
- 8. At the threshold point where the secondary emission sustains the plasma and the Townsend Avalanche is formed, Equations 6 and 7 are balanced and Γi(0) can be substituted, leading to: Γe(0) γ = Γe(0)(e αd − 1)….(8) 1/γ = e αd − 1…(9) αd = ln (1 + 1/γ)…(10)
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