Advanced Solid State Physics

with eqn. (219):

⇒χ(ω) =

σ(ω)

iω 0

=

ne^2

iω 0 m

(πδ(ω) +

i

ω

) (226)

We used:

x(ω,t) = x(ω)e−iωt⇒v(ω,t) = x(ω)e−iωt(−iω)⇒x(ω) =

v(ω,t)

−iω

eiωt =

v(ω)

−iω

So the dielectric constant is derived:

(ω) = 1 +χ(ω) = 1 +

ne^2

iω 0 m

(πδ(ω)−

i

ω

) = 1−

ωp^2

ω^2

+iωp^2 πδ(ω) (227)

with the Plasma frequency:

ω^2 p=

ne^2

m 0

Fig. 108 shows the real part of the dielectric constant:

(ω) = 1−

ω^2 p

ω^2

(228)

At the Plasma frequencyωpthe dielectric constant is zero. Below zero (so below the Plasma frequency)
the material reflects and above zero (above the Plasma frequency) the material transmits incoming
waves.

Figure 108: Real part of the dielectric constant of a collisionless material. Below the plasma frequency
the material reflects, above the plasma frequency the material transmits incoming waves.