Langmuir Sheaths

Let us construct a one-dimensional model of an unmagnetized, steady-state, Langmuir sheath. Suppose that the wall lies at , and that the plasma occupies the region . Let us treat the ions and the electrons inside the sheath as collisionless fluids. The ion and electron equations of motion are thus written

respectively. Here, is the electrostatic potential. Moreover, we have assumed uniform ion and electron temperatures, and , respectively, for the sake of simplicity. We have also neglected any off-diagonal terms in the ion and electron stress-tensors, because these terms are comparatively small. Note that quasi-neutrality does not apply inside the sheath, and so the ion and electron number densities, and , respectively, are not necessarily equal to one another.Consider the ion fluid. Let us assume that the mean ion velocity, , is much greater than the ion thermal velocity, . Because, as will become apparent, , this ordering necessarily implies that : that is, that the ions are cold with respect to the electrons. It turns out that plasmas in the immediate vicinity of solid walls often have comparatively cold ions, so our ordering assumption is fairly reasonable. In the cold ion limit, the pressure term in Equation (4.243) is negligible, and the equation can be integrated to give

(4.245) |

(4.246) |

(4.247) | ||

(4.248) |

Consider the electron fluid. Let us assume that the mean electron velocity, , is much less than the electron thermal velocity, . In fact, this must be the case, otherwise, the electron flux to the wall would greatly exceed the ion flux. Now, if the electron fluid is essentially stationary then the left-hand side of Equation (4.244) is negligible, and the equation can be integrated to give

Here, we have made use of the fact that at the edge of the sheath.Poisson's equation is written

(4.250) |

(4.252) |

(4.253) |

(4.255) |

The ion current density at the wall is

(4.257) |

(4.258) |

(4.259) |

(4.260) |

For a hydrogen plasma, . Thus, hydrogen ions enter the sheath with an initial energy , fall through the sheath potential, and so impact the wall with energy .

Combining Equations (4.247)–(4.249), (4.254), (4.262), and making use of the constraint , we arrive at the following set of equations that characterize the structure of a Langmuir sheath:

Equation (4.263) can be solved numerically, subject to the boundary condition (4.264), to give the results summarized in Figure 4.4. The figure illustrates how the deviation from quasi-neutrality within the sheath generates an electric potential that greatly reduces the electron number density at the wall, and also accelerates the ions as they pass through the sheath (toward the wall).