Advanced Solid State Physics

HerePrepresents thePrinciple part, which is important because at a particular pointω′ = ωthe
integral gets undefined (singularity). With this Principle part it is possible to integrate around this
singularity.

13.1 Optical Properties of Materials

Dielectric Constant
One way to calculate the dielectric functions is to measure the index of refractionnand the extinction

coefficientK(this coefficient tells how quickly a wave dies out). Taking the square of the following
relationship
√
μ 0  0 (ω) = n(ω) +iK(ω)

gives the dielectric function:

(ω) =

n^2 −K^2 + 2inK

μ 0  0

= ′(ω) +i′′(ω) (213)

Important to mention is thatn^2 −K^2 and 2 nKare related by the Kramers-Kronig relation.

Dielectric Response of Insulators
Insulators do not have free electrons, which means that their charge is bound. The electrons can be
modeled to be coupled by a spring to the ions and as a result it’s possible to describe this function as
a harmonic oscillator. A hit of this function with aδ-function will give the response function (Green’s
function) for this linear harmonic oscillator:

m

d^2 g

dt^2

+b

dg

dt

+kg = δ(t)

g(t) = −

1

b

exp

(−bt

2 m

)

sin

(√

4 mk−b^2

2 m

t

)

t > 0

To have the same notation as Kittel, the response functiong(t)is calledα(t)now.

Response of a Harmonic Oscillator
The solutions (assumption) for the displacement and the force have the formx(ω)e−iωtandF(ω)e−iωt.

m

d^2 x

dt^2

+b

dx

dt

+kx = F(t)

=⇒

−ω^2 mx(ω)−iωbx(ω) +kx(ω) = F(ω)

The generalized susceptibilityα(ω)is

α =

F(ω)

x(ω)

=

F

−ω^2 m−iωb+k

=

F

m

ω 02 −ω^2 +iωb

(ω^20 −ω^2 )^2 +ω^2 b^2

(214)