Thermal properties of Solids: phonons w k p/a

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

Longitudinal vibration of a linear chain of two masses

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

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 -3000 K, Cu -320K, Pb -90K

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

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