Advanced Solid State Physics

can solve this system of equations by calculating
∣∣
∣∣
∣

μ 0 ε 0 ω^2 −k^2 μ 0 ω^2

Ne^2 / m ω^2 −ω^2 T

∣∣

∣∣

∣= 0, (133)

which gives us a new dispersion relationship betweenkandωthat is shown in fig. 74(b). There are
two solutions for everyk, one for the upper branch and one for the lower branch. In the frequency
range betweenωTandωLthere are no waves at all allowed in the crystal. This means that light Bragg
reflects off the sound wave and sound Bragg reflects off the light wave.

In fig. 75 you can see the reflectance of NaCl. At the frequency range of the gap in the polariton
dispersion relationship the reflectance of NaCl is very high as expected. The dielectric constant as a

Figure 75: Reflectance of NaCl; gap = frequency gap of polaritons.

function ofωnow looks like

ε(ω) = 1 +

P

ε 0 E

= 1 +

Ne^2

m(ωT^2 −ω^2 )

=

ω^2 Tε(0)−ω^2 ε(inf)

ω^2 T−ω^2

(134)

and atω=ωT there is divergence as you can see in fig. 76. Therefore there is a region where the
dielectric constant is negative which means that the wave vectorkbecomes imaginary. This indicates
that waves decay exponentially and are reflected out of the material. So the region whereε < 0
corresponds to the frequency gap of the polaritons.

11.3.3 Magnons

Magnons are the low lying excitations of the ordered ferro- (or antiferro) magnetic state. The ground-
state (zero temperature) of a ferromagnetic linear chain is visualized in figure 77 (a). Let us calculate
the exchange energy in the Heisenberg model

Hˆexc=− 2 J

N∑− 1

k=1

Sˆ~

k·

S~ˆ

k+1 (135)