Fundamentals of Plasma Physics

2.7 Sheath physics and Langmuir probe theory 57

and so the electron current collected at the probe is

Ie=−n 0 eA

√

κTe

2 πme

e−e|

̄φ(x)|/κTe

. (2.120)

Thus, the combined electron and ion current collected by the probe is

I = Ii+Ie (2.121)

= n 0 cseA−n 0 eA

√

κTe

2 πme

e−e|

̄φ(x)|/κTe

.

The electron and ion currents cancel each other when

√

2 κTe

mi

=

√

κTe

2 πme

e−e|

φ ̄(x)|/κTe

(2.122)

i.e., when

e| ̄φprobe|/κTe = ln

√

mi

4 πme

= 2. 5 for hydrogen. (2.123)

This can be expressed as

φprobe =φplasma−

κTe

e

ln

√

mi

4 πme

(2.124)

and shows that when the probe potential is more negative than the plasma potential by
an amount Te ln

√

mi/ 4 πmewhereTeis expressed in electron volts, then no current
flows to the probe. This potential is called thefloating potential, because an insulated
object immersed in the plasma will always charge up until it reaches thefloating potential
at which point no net currentflows to the object.
These relationships can be used as simple diagnostic for the plasma density and tem-
perature. If a probe is biased with a large negative potential, then no electrons are collected
but an ionflux is collected. The collected current is called the ion saturation currentand
is given byIsat=n 0 cseA.The ion saturation current is then subtracted from all subse-
quent measurements giving the electron currentIe=I−Isat=n 0 eA(κTe/ 2 πme)^1 /^2
exp

(

−e| ̄φ(x)|/κTe

)

.The slope of a logarithmic plot ofIeversusφgives 1 /κTeand can
be used to measure the electron temperature. Once the electron temperature is known,cs
can be calculated. The plasma density can then be calculated from the ion saturation cur-
rent measurement and knowledge of the probe area. Langmuir probe measurements are
simple to implement but are not very precise, typically having an uncertainty of a factor of
two or more.