LANGMUIR WAVES - THE BOHM-GROSS DISPERSION RELATION

and so on for v 1 and E 1 . From here, it is easy to see that we can make the substitutions
∂
→ −iω, ∇ → ik (11)
∂t
It is easiest to perform this substitution on Gauss’s Law to obtain an expression for E 1 :
∇ · E 1 = − en e1
ɛ 0
With these substitutions, equations 8 and 9 become
Plugging equation 13 into equation 12:
→ ik · E 1 = − en e1
ɛ 0
(12)
−iωn e1 + ik · n e0 v 1 = 0 ⇒ n e1 = n e0
ω k · v 1 (13)
Dotting both sides of equation 14 with k and rearranging:
And putting this into equation 15:
and finally putting in equation 12
−iωmn e0 v 1 = −en e0 E 1 − iγk B T kn e1 (14)
k · v 1 =
ik · E 1 = −e n e0
ωɛ 0
k · v 1 (15)
en e0
iωmn e0
k · E 1 + γk BT n e1
ωmn e0
k 2 (16)
ik · E 1 = −e n e0
ωmɛ 0
en e0
iωn e0
k · E 1 − e n e0
ωɛ 0
γk B T n e1
ωmn e0
k 2 (17)
− en e1
ɛ 0
This is really ugly. Let’s clean this up
Multiplying through to solve for the frequency ω:
= −e n e0 en e0 en e1
− e n e0 γk B T n e1
k 2 (18)
ωmɛ 0 ωn e0 ɛ 0 ωɛ 0 ωmn e0
1 = n e0e 2
ω 2 mɛ 0
+ γk BT
ω 2 m k2 (19)
ω 2 = n e0e 2
mɛ 0
+ γk BT
m k2 ✷ (20)
This is the so-called Bohm-Gross Dispersion Relation. If we define the electron plasma frequency
ω 2 pe = n e0e 2
mɛ 0
(21)
and the electron thermal velocity
vth,e 2 = k BT
m
then the Bohm-Gross relationship can be written
(22)
ω 2 = ω 2 pe + γv 2 th,ek 2 ✷ (23)
Note, in particular, that for ‘cold’ electrons, the frequency of oscillations reduces to the plasma
frequency. If the electrons are cold, then the group velocity ∂ω/∂k = 0, so the wave does not
propagate. A nonzero electron temperature yields a nonzero group velocity, so the wave will
propagate.