Townsend’s Current Growth Equation (Considering Primary and Secondary Ionization Both)

I = I₀ e^αd [Considering only Primary ionization]

I = I₀ e^αd / [1 – γ (e^αd – 1)]  [Considering both Primary and secondary ionization]

Let n₀  be the number of electrons released from cathode due to UV radiations.

n₊ – be the number of electrons released from cathode due to the positive ion bombardment.

n_d  – number of electrons reaching anode.

γ = Townsends secondary ionization coefficient i.e. number of electrons released from cathode per incident positive ion, photon or metastable.

We have equation from Townsends primary ionizaton as,

n_d = n₀ e^αd …. (1)

The equation changes to

n_d = (n₀ + n₊) e^αd ….(2)

Here n₀ + n₊ – are the number of electrons at the cathode,

n₊ = (additional electron) γ

n₊ = [ n_d – (n₀ +
n₊)]y 

Note : Solve bracket, take n₊ on L.H.S. remaining part on R.H.S., put  n₊ value in Equation (1) and (2), simplify and get the value for n_d’  n_d / time is current equation.

n₊ = n_d . y – n₀ .γ – n₊
.γ

n₊ + n₊ .γ = n_d . y – n₀
.γ

n₊ (1 + γ) = n_d . y – n₀ .γ

n₊ = y  (n_d – n₀) / (1 + γ)

Put value of n₊ in Eqn (2)

n_d = [n₀ + γ  (n_d – n₀) / (1 + γ)] e^αd

[n₀ + n₀.y + n_d^.y-
n₀.y / (1 + γ)] e^αd

n_d = [n₀ + n_d^.γ / (1 + γ)] e^αd

n_d (1 + γ) = n₀ + n_d γ

n_d + n_d^.γ = n₀. e^αd + n_d^.γ.
e^αd

n_d + n_d^.y – n_d^.γ.
e^αd = n₀. e^αd

n_d (1 + y – γ. e^αd) = n₀. e^αd

n_d = n₀. e^αd / [1 + γ (1 – e^αd)]

Divided by time,

n_d / t = = n₀. e^αd / [1 + γ (1 – e^αd)] /t

I = I₀ . e^αd/ 1 – y (e^αd – 1)

This is Townsend’s current grown equation due to (α) primary and secondary (γ) ionization.