this is used in stained glass^11. In order to make red glass, gold particles are added during the production
process. The surface plasma waves of these gold particles are down in the red so they have the right
frequency to scatter red light and make the glass look red.
Surface plasmons are also used for biosensors. E.g. an organic light emitting diode is put on a gold
layer and it emits light that excites surface plasmons on the gold structure. On this gold layer there
are aptamers. Aptamers are molecules that stick with one side to a metal and with the other side to
some chemical or biological material. If molecules are attached to the aptamers (see fig. 81(b)), the
dielectric constant right at the interface changes and therefore the wavelength of the surface plasmons
changes too (the frequency is bound to the OLED). This change can be detected.
11.3.6 Frenkel Excitations
11.4 Translational Symmetry
Quasiparticles can be labeled with the symmetry labelk. The translation operatorT shifts a function
one period (a...lattice constant):
Tf(x) =f(x+a) (171)Solutions of Bloch form are eigenfunctions of the translation operator.
ψk(x) = eikxuk (172)=⇒
Tψk(x) = eik(x+a)uk(x+a) = eikxeikauk(x) = eikaψk(x) (173)ukis a periodic function souk(x) = uk(x+a). It is easy to see thateikais the eigenvalue of the
translation opertator.
The solution depends only onk(becauseais always the same) so the labeling of the states happens
only in terms ofk.
The translation operator commutes with the Hamiltonian even when the interaction, the symmetry of
the crystal, is the same. As a result it is possible to diagonalize the eigenfunctions of the Hamiltonian
and the translation operator. Furthermore if you know thekyou can also specify the eigenstates of the
energy. So thequantum numbers which are used, come from the symmetry(e.g. Hydrogen
atom).
Very important is that there isn’t a good correspondence between thekand the momentum anymore,
cause they don’t describe the same.
One example of a quasiparticle system would be plasmons. Plasmons are oscillating electric charges
which build a slow wave. In fact there are millions or billions of electrons which are involved in this
wave, but (!) it’s still a quasiparticle system with a specifickvector because millions of electrons
making an interaction electron system.
(^11) Buntglas