Analytical approximations for the sheath potential profile

The space charge sheath in front of a plane negative wall is frequently described with rather poor approximations since Poisson's equation cannot be solved analytically. Accounting for the correct boundary conditions at the sheath edge, convenient analytical approximations are constructed both in the fluid and in the kinetic approach. The fluid solution is universal and does not depend on the special plasma properties. In the kinetic analysis, in contrast, different plasma models must be distinguished because the space charge formation depends on the ion velocity distribution at the sheath edge. The differences are significant in particular at low sheath potentials. Explicit formulae are given for the Tonks-Langmuir model and for the charge exchange model. All approximations represent the sheath potential profiles with surprising accuracy.