LATTICE VIBRATION IV
Real crystal system
The Number Of Branches
Phonons
Simple Crystal Systems w vs k curves
real crystals are more complicated
Force constant may vary - the branches are not degenerate
the atoms vibrate in three dimensions with two vibration modes and three vibrational branches, one longitudinal and two transverse
1D case
only nearest-neighbor interactions.
eq of motion 
solution 
3D case
to avoid mathematical details we shall present only a qualitative discussion
first, the monatomic Bravais lattice, in which each unit cell has a single atom
The solution of this equation in 3D can be represented in terms of normal modes:
when substitute, we obtain three simultaneous equations
The roots of this equation lead to three different dispersion relations, or three dispersion curves
non-Bravais three-dimensional lattice
the unit cell contains two or more atoms.
three branches are acoustic, and the remaining (3z − 3) are optical
lattice vibration for the monatomic chain (z = 1)
For diatoms, z = 2
Pb
Na
Since z = 1 three branches are expected in any one k-direction.
The particular directions shown in the figure are high-symmetry ones so the transverse modes are degenerate
FCC crystal structure
BCC crystal structure
Since z = 1, again we expect the same number of modes as in the Pb case
However, in the [110] direction the transverse branches are no longer degenerate
Nevertheless, the branches have a shape similar to those shown in longitudinal waves of a linear chain
.
phonon momentum
photons
phonons
quanta of lattice vibrations
energies of phonons are quantized
Quanta of electromagnetic radiation
Energies of photons are quantized as well
A phonon of wavevector k will interact with particles such as photons, neutrons and electrons as if it had a momentum ħk
However, phonon does not carry physical momentum
The reason is that the center of mass of the crystal does not change it position under vibrations (except k=0).
phonon generation
Piezoelectric
Thermal Excitation
Electron Tunneling