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 (4.254) is negligible, and the equation can be integrated to give

(4.256) |

Here, and are the mean ion velocity and electrostatic potential, respectively, at the edge of the sheath (i.e., ). Now, ion fluid continuity requires that

(4.257) |

where is the ion number density at the sheath boundary. Incidentally, because we expect quasi-neutrality to hold in the plasma outside the sheath, the electron number density at the edge of the sheath must also be (assuming singly-charged ions). The previous two equations can be combined to give

(4.258) | ||

(4.259) |

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 (4.255) 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

(4.261) |

It follows that

Let , , and

(4.263) |

where is the Debye length. Equation (4.262) transforms to

(4.264) |

subject to the boundary condition as . Multiplying through by , integrating with respect to , and making use of the boundary condition, we obtain

Unfortunately, the previous equation is highly nonlinear, and can only be solved numerically. However, it is not necessary to attempt this to see that a physical solution can only exist if the right-hand side of the equation is positive for all . Consider the limit . It follows from the boundary condition that . Expanding the right-hand side of Equation (4.265) in powers of , we find that the zeroth- and first-order terms cancel, and we are left with

(4.266) |

Now, the purpose of the sheath is to shield the plasma from the wall potential. It can be seen, from the previous expression, that the physical solution with maximum possible shielding corresponds to , because this choice eliminates the first term on the right-hand side (thereby making as small as possible at large ) leaving the much smaller, but positive (note that is positive), second term. Hence, we conclude that

This result is known as the

The ion current density at the wall is

(4.268) |

This current density is negative because the ions are moving in the negative -direction. What about the electron current density? Well, the number density of electrons at the wall is , where is the wall potential. Let us assume that the electrons have a Maxwellian velocity distribution peaked at zero velocity (because the electron fluid velocity is much less than the electron thermal velocity). It follows that half of the electrons at are moving in the negative- direction, and half in the positive- direction. Of course, the former electrons hit the wall, and thereby constitute an electron current to the wall. This current is , where the comes from averaging over solid angle, and is the mean electron speed corresponding to a Maxwellian velocity distribution (Reif 1965). Thus, the electron current density at the wall is

(4.269) |

In order to replace the electrons lost to the wall, the electrons must have a mean velocity

(4.270) |

at the edge of the sheath. However, we previously assumed that any electron fluid velocity was much less than the electron thermal velocity, . As is clear from the previous equation, this is only possible provided that

(4.271) |

that is, provided that the wall potential is sufficiently negative to strongly reduce the electron number density at the wall. The net current density at the wall is

Of course, we require in a steady-state sheath, in order to prevent wall charging, and so we obtain

(4.273) |

We conclude that, in a steady-state sheath, the wall is biased negatively with respect to the sheath edge by an amount that is proportional to the electron temperature.

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 .

A *Langmuir probe* is a device used to determine the electron temperature and electron number density of a plasma. It works by inserting an electrode that is biased with respect to the vacuum vessel into the plasma. Provided that the bias voltage is not too positive, we would
expect the probe current to vary as

where is the surface area of the probe, and its bias with respect to the vacuum vessel. [See Equation (4.272).] For strongly negative biases, the probe current saturates in the ion (negative) direction. The characteristic current that flows in this situation is called the

For less negative biases, the current-voltage relation of the probe has the general form

(4.276) |

where is a constant. Thus, a plot of versus gives a straight-line from whose slope the electron temperature can be deduced. Note, however, that if the bias voltage becomes too positive then electrons cease to be effectively repelled from the probe surface, and the current-voltage relation (4.274) breaks down. Given the electron temperature, a measurement of the ion saturation current allows the electron number density at the sheath edge, , to be calculated from Equation (4.275). Now, in order to accelerate ions to the Bohm velocity, the potential drop across the pre-sheath needs to be , where is the electric potential in the interior of the plasma. It follows from Equation (4.260) that the relationship between the electron number density at the sheath boundary, , and the number density in the interior of the plasma, , is

(4.277) |

Thus, can also be determined from the probe.