The plasma is kept inside a vacuum chamber that is a finite size in all practical plasma systems.

Think about a 1-D model where the plasma has no magnetic field and no discernible electric field. Potential then becomes zero inside. Ion and electron are combined and lost as they strike the wall. Due to their far higher thermal velocities than ions, electrons have lost energy more quickly, leaving the plasma with a net positive charge. If the wall is positive and w is negative, the plasma must therefore have a potential positive in relation to the wall. Since Debye Shielding will limit the potential variation to a layer several Debye lengths in thickness, this potential cannot be diffused across the entire plasma. This layer, which must exist on all cold walls with which plasma is in contact is called sheath.

The height of the potential barrier adjust itself so that the flux of electrons that have enough energy to go over the barrier to the wall is just equal to flux of ions reacting the wall.

The function of sheath is to form a potential barrier so that mobile (electrons) species is confined electrostatically.

The plasma sheath equation

To get exact behaviour of potential 𝜙 we must treat non linear problem. At the plane x=0, ions are imagined to enter the sheath region from main plasma with the drift velocity Vo. Let us assume Ti=0 so that all ions have velocity uo at x=0.

The potential 𝜙(r) is assumed to decrease monotonically with x. If u(x) is the ion velocity, then conservation of energy is

1/2 mu² + e 𝜙(x) = 1/2 mu_o²

The ion equation of continuity then given ion density ni in terms of density n_o in main plasma

For steady state

In steady state, the electrons will follow Boltzmann distribution

from Poisson's equation

∇E = - e/𝜖 o (ne - ni)

Multiplying equation 4 by e/KTe

We have,

This is non linear equation of a plasma sheath and it has an acceptable solution only if M is large enough.

Solving these equations provides insights into the electric potential, charge density distribution, and ion and electron dynamics within the sheath. The specific form and complexity of the equations may vary depending on the assumptions and additional effects considered in the particular plasma sheath model.

This note is a part of the Physics Repository.

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