Saha equation
n2
i
nn
= 2.405 · 1021
T3/2
e
− U
kBT
(1)
Common unit conversions in plasma physics
1 eV ≈ 1.602 · 10−19 J
1 eV = e
k K ≈ 11604 K 1
8000 K of temperature is only about 0.6894 eV of energy.
1
This just a unit conversion. A particle with temperature T will, generally,
have a different mean energy (based on the particle’s degrees of freedom) ->
Use this with caution!
Ideal gas law
The gas consists of neutral and charged particles. Assume charged
particles are positive ions and electrons. Assume all species have the
same temperature T.
Then the ideal gas law is:
p = pe + pi + pn =
kT
V
(Ne + Ni + Nn) = kT(ne + ni + nn) (2)
To maintain overall neutrality:
ne + ni = 2ni (3)
Combine every equation. . .
n2
i
nn
= 2.405 · 1021
T3/2
e
− U
kBT
≡ A (4)
nn =
p
kT
− 2ni ≡ b − 2ni (5)
(6)
n2
i + 2Ani − Ab = 0 (7)
Figure 1: wolframalpha.com can solve this
In numbers..
For T = 300 K:
b = 2.5 · 1025
(8)
A = 3.2 · 10−219
(9)
ni = 2.8 · 10−97
(10)
For T = 8000 K:
b = 9.2 · 1023
(11)
A = 1.3 · 1018
(12)
ni = 1.1 · 1021
(13)
Degree of ionization
For T = 300 K:
ni
nn + ni
= 1.2 · 10−122
(14)
For T = 8000 K:
ni
nn + ni
= 0.0012 (15)