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modes

Hooke’s law: Vibration frequency   f = force constant, M = mass. modes. Atomic Vibrations in Crystals = Phonons. 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.

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modes

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  1. Hooke’s law: Vibration frequency f = force constant, M = mass modes Atomic Vibrations in Crystals = Phonons 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.

  2. Transverse modes (Oscillating Dipole) r

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

  4. Harmonic Anharmonic T>0 T=0 a Anharmonic oscillator and thermal expansion 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.

  5. Bragg reflection makes neutrons (and X-rays) monochromatic. Triple-axis spectrometer: k E0E Measuring phonons by inelastic (E≠0)neutron scattering Energy and momentum conservation: E = E0 Ephon k =k0 kphon+ Ghkl E,k E0,k0 Ephon,kphon

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

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

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