16.522, Space Propulsion
Prof. Manuel MartinezSanchez
Lecture 10: Electric Propulsion  Some Generalities on Plasma (and Arcjet
Engines)
Ionization and Conduction in a Highpressure Plasma
A normal gas at T <3000K is a good electrical insulator, because there are almost no
∼
free electrons in it. For pressure >
∼
0.1 atm, collision among molecule and other
particles are frequent enough that we can assume local Thermodynamic Equilibrium,
and in particular, ionizationrecombination reactions are governed by the Law of
Mass Action. Consider neutral atoms (n) which ionize singly to ions (i) and electrons
(e):
n
R
e +i
(1)
One form of the Law of Mass Action (in terms of number densities n=
P
j
kT
, where T
j
is the same for all species) is
nn
ei
()
(2)
=S T
n
n
Where the “Saha function” S is given (according to Statistical Mechanics) as
3
()
ST
i
q
n
= 2
q
⎛
π
e
⎜
h
2
⎟
2m
kT
⎞
⎝
⎠
eV
i
e
kT

(3)
q
i
= Ground state degeneracy of ion (= 1 for H
+
)
q
n
= Ground state degeneracy of neutral (= 2 for H)
m
e
= mass of electron = 0.91 × 10
30
Kg
k = Boltzmann constant = 1.38 ×10
23
J/K
(Note: k = R/Avogadro’s number)
h = Plank’s constant = 6.62x10
34
J.s.
V
i
= Ionization potential of the atom (volts)
(V
i
= 13.6 V for H)
Except for very narrow “sheaths” near walls, plasmas are quasineutral:
e
n=
n
i
(4)
So that
e
n
n
n
()
=S T
(3’)
can be used.
2
16.522, Space Propulsion
Lecture 10
Prof. Manuel MartinezSanchez
Page 1 of 12
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e
to n
n
. A second relation is needed and very often it is a
specification of the overall pressure
P
= (
n
+
n
i
+
n
)
kT =
(
2n
+
n
)
kT
(5)
e
n
e
n
Combining (3’) and (5),
2
()
⎛
⎜
P
2n
e
⎟
=
ST
n
=
ST
⎝
kT
⎞
()
(
n2n
e
)
e
⎠
P
Where n=
is the total member density of all particles.
kT
We then have
2
n+
2S
n
S
n
=
0
e
e
S
2
(6)
n=

S
+ +
S
n
=
n
e
1+ 1 +
n
ST
()
Since S increases very rapidly with T, the limits of (6) are
→
S
n
(
→
0
)
(Weak ionization)
n
⎯⎯⎯⎯⎯
e
T0
T0
→
→
⎯⎯⎯⎯⎯
n
n
e
T
→∞
→
2
(Full ionization)
G
Once an electron population exists, an electric field Ewill drive a current density
G
j through the plasma. To understand this quantitatively, consider the momentum
balance of a “typical” electron. It sees an electrostatic force
G
G
F=

e
E
(7)
E
It also sees a “frictional” force due to transfer of momentum each time it collides
with some other particle (neutral or ion). Collisions with other electrons are not
counted, because the momentum transfer is in that case internal to the electron
species. The ions and neutrals are almost at rest compared to the fastmoving
electrons, and we define an effective collision as one in which the electron’s directed
momentum is fully given up. Suppose there are
ν
e
of these collisions per second
(
ν
e
=collision frequency per electron). The electron loses momentum at a rate
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 Fall '05
 ManuelMartinezSanchez
 Electron, Propulsion, Prof. Manuel MartinezSanchez, Space Propulsion

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