Fundamentals of Plasma Physics

54 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD

metal wires used to diagnose low temperature plasmas. Biasing a Langmuir probe at a se-

quence of voltages and then measuring the resulting current provides a simple way to gauge

both the plasma density and the electron temperature.

x
 0

plasm a

Langmuir probe

(or metalwall)

probe

−x 0

potentialwith convex curvature

plasma sheath

Figure 2.6: Sketch of sheath. Ions are accelerated in sheath to probe (wall) atx= 0whereas
electrons are repeled. Convex curvature of sheath requiresni(x)to always be greater than
ne(x).

The model presented here is the simplest possible model for sheaths and Langmuir

probes and is one-dimensional. The geometry, sketched in Fig.2.6, idealizes the Langmuir

probe as a metal wall located atx= 0and biased to a potentialφprobe.;this geometry

could also be used to describe an actual biased metal wall atx= 0 in a two-dimensional

plasma. The plasma is assumed to be collisionless and unmagnetized and tohave an am-

bipolar potentialφplasmawhich differs from the laboratory reference potential (so-called

ground potential) because of a difference in the diffusion rates of electrons and ions out

of the plasma. The plasma is assumed to extend into the semi-infinite left-hand half-plane

−∞< x < 0 .Ifφprobe=φplasma, then neither electrons nor ions will be accelerated or

decelerated on leaving the plasma and so each species will strikethe probe (or wall) at a

rate given by its respective thermal velocity. Sinceme<<mi,the electron thermal veloc-

ity greatly exceeds the ion thermal velocity. Thus, forφprobe=φplasma the electronflux

to the probe (or wall) greatly exceeds the ionflux and so the current collected by the probe

(or wall) will be negative.

Now consider what happens to this electronflow if the probe (or wall) is biasednegative

with respect to the plasma as shown in Fig.2.6. To simplify the notation, a bar will be used