Download Phonons

Atomic Vibrations in Crystals = Phonons
Hooke’s law: Vibration frequency  
f
M
f = force constant, M = mass
Test for phonon effects by using isotopes with different mass, for example in superconductivity, where electron pairs are formed by the electron-phonon interaction.
modes
Transverse modes
(Oscillating Dipole)
r
Classical vs. quantum vibrations in a molecule
r
r
Quantum
probability
Classical
probability
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 …
Harmonic
Anharmonic
T>0
T=0
a
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 harmonic oscillator does not produce
such a shift.
Measuring phonons by inelastic (E≠0) neutron scattering
E,k
Energy and momentum conservation:
E = E0  Ephon
E0,k0
k = k0  kphon + Ghkl
Ephon, kphon
Bragg reflection makes neutrons (and X-rays) monochromatic.
Triple-axis spectrometer:
k
E0
E
Measuring phonons by inelastic photon scattering
(Raman Spectroscopy)
Tphonon
photon
phonon
Tphoton
The phonon wave modulates the
light wave, creating side bands
(like AM radio).
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.