2. modes Atomic Vibrations in Crystals = Phonons Hooke’s law: Vibration frequency   f = force constant, M = mass Test for phonon effects by using isotopes with different mass, for example in super- conductivity, where electron pairs are formed by the electron-phonon interaction.

3. rr Transverse modes (Oscillating Dipole)

4. Classical vs. quantum vibrations in a molecule Classical probability Quantum probability rr rr

5. Anharmonic oscillator and thermal expansion Anharmonic T=0 T>0 Harmonic a A realistic potential energy curve between two atoms is asymmetric: short-range Pauli repulsion versus long-range Coulomb attraction (see Lect. 5, p. 4): U(r)  (  r) 2  (  r) 3 … This asymmetry causes anharmonic oscillations. The probability density |  | 2 shifts towards larger r for the higher vibrational levels. These are excited at higher temperature. The symmetric potential of the har- monic oscillator does not produce such a shift.

6. Measuring phonons by inelastic (  E ≠ 0) neutron scattering E0,k0E0,k0 E,k E phon, k phon Energy and momentum conservation: E = E 0  E phon k = k 0  k phon + G hkl Bragg reflection makes neutrons (and X-rays) monochromatic. Triple-axis spectrometer: E 0 E k

7. Measuring phonons by inelastic photon scattering (Raman Spectroscopy) T photon T phonon  phonon  photon The phonon wave modulates the light wave, creating side bands (like AM radio).

8. Measuring phonons by inelastic electron scattering Electrons interact very strongly with optical phonons in ionic solids. That gives rise to multiple phonon losses. Electron Energy Loss Spectroscopy (EELS) Probing Depth: Neutrons: cm Photons:  m-cm Electrons: nm 