1. Thermal properties of Solids: phononsw k
p/a

2. Brillouin zone First Brillouin zone is range of wave vectors k: k2 k1
p/a
p/a k
Displacement of lattice point
First Brillouin zone is range of
wave vectors k:
-p/a£ k £ p/a

3. Longitudinal vibration of a linear chain of two masses

4. Vibration of a linear chain of two masses
optical w2 w1
acoustic
-2p/a k
2p/a
1st critical frequency: heavy masses only moving,
light masses at nodes
2st critical frequency: light masses only moving,
heavy masses at nodes
3st critical frequency: light and heavy masses are
moving in opposite directions

5. Quantized lattice vibrations
Quantized model of the crystal vibrations: there is set of 3N independent
linear oscillators( modes) with energy E=(n(w)+1/2) hw
The mean number of phonons in the mode with w is
Debye’s frequency wD : the largest frequency of vibration in crystal assuming
linear dispersion: w = v k. Debye’s temperature Q= hwD/kB
Typical acoustic phonon frequency is ~1013 Hz, optical phonon frequency
~ 1014 Hz, Deby’s temperature: diamond K, Cu -320K, Pb -90K

6. Inelastic neutron scattering
Neutrons can be scattered from the crystal away from Bragg’s law
when absorbs or emits phonon:

7. Infrared absorption in ionic crystals
Transmitted light in infrared range, w~ 1014 Hz (l~40-100mm)
are absorbed by ionic crystals with optical mode of phonons
Transmittance through thin
film (0.17mm) of NaCl Cl Cl Cl Na Na 50 60 70
l(mm)
Transmittance
100%
Ions of Cl and Na are moving in opposite
directions