Page 34 - Electrical Properties of Materials

Plasma waves 17

If B = 0, then eqn (1.22) takes the simple form
∂E E E
J + = 0. (1.65)
∂t

We need the equation of motion, which for longitudinal motion will have
exactly the same form as for transverse motion, namely

∂v
m = eE E E , (1.66)
∂t
∗ Ignoring losses will considerably re-
strict the applicability of the formulae
where we have neglected the damping term mv/τ. ∗
derived, but our aim here is to show no
Current density and velocity are related again by more than the simplest possible case.
ev. (1.67) N e 0 is the equilibrium density of
J = N e 0
electrons.
We have changed over to scalar quantities.
Substituting E E E from eqn (1.66) and J from eqn (1.67) into (1.65), we obtain

2
m ∂ v
ev + = 0. (1.68)
N e 0 2
e ∂t
Following again our favourite method of replacing ∂/∂t by –iω, eqn (1.69)
reduces to
m 2

e + (–ω ) = 0. (1.69)
v N e 0
e
Since v must be finite, this means
m 2
e – ω = 0, (1.70)
N e 0 v
e
(b)
Light line
or, rearranging,
e 2 (a)
2
N e 0
ω = . (1.71)
m v p
This is our dispersion equation. It is a rather odd one because k does not appear v p
in it. A relationship between k and ω gives the allowed values of k for a given √ 2
(c)
ω.If k does not appear in the dispersion equation, all values of k are allowed.
On the other hand, there is only a single value of ω allowed. Looking at it more
carefully, we may recognize that it is nothing else but ω p , the frequency we met
previously as the critical frequency of transparency for electromagnetic waves.
k
Historically, it was first discovered in plasma oscillations (in gas discharges by
Langmuir); so it is more usual to call it the ‘plasma frequency’, and that is Fig. 1.8
where the subscript p comes from. Dispersion curves of plasma waves.
The dispersion curve given by eqn (1.71) is just a straight horizontal line, (a) Plasma density wave, (b) bulk
as shown in Fig. 1.8. The dispersion curve of the electromagnetic wave cor- plasma wave or bulk plasmon
responding to eqn (1.53) may also be seen in the same figure. As explained polariton, (c) surface plasma wave or
before, in the latter case there is no propagation unless the frequency is above surface plasmon polariton. The
equation of the light line is ω = kc.
the plasma frequency. For high enough frequencies the dispersion curve tends