Thermal properties of Solids: phonons

1. a m n n+1 n-1 p/a p/a 0 k Thermal properties of Solids:phonons w

2. k2 k1 Brillouin zone k1 k2 w p/a p/a 0 k Displacement of lattice point First Brillouin zone is range of wave vectors k: -p/a£ k £ p/a

3. m m M M 2n 2n+1 2n+2 2n-1 a/2 a/2 Longitudinal vibration of a linear chain of two masses

4. Vibration of a linear chain of two masses w w3 optical w2 w1 acoustic k 0 -2p/a 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 wis 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

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

7. 100% Cl Cl Cl Na Na Transmittance 70 50 60 l(mm) 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 Ions of Cl and Na are moving in opposite directions