1. 4. Phonons Crystal Vibrations
Vibrations of Crystals with Monatomic Basis
Two Atoms per Primitive Basis
Quantization of Elastic Waves
Phonon Momentum
Inelastic Scattering by Phonons

2. Harmonic approximation: quadratic hamiltonian : elementary excitations
Electrons, polarons & excitons are quasi-particles

3. Vibrations of Crystals with Monatomic Basis
First Brillouin Zone
Group Velocity
Long Wavelength Limit
Derivation of Force Constants from Experiment

4. Entire plane of atoms moving in phase → 1-D problem
Force on sth plane =
(only neighboring planes interact )
Equation of motion: → →
Dispersion relation

5. Propagation along high symmetry directions → 1-D problem
E.g. , [100], [110], [111] in sc lattice.
longitudinal wave
transverse wave

6. First Brillouin Zone → Only K  1st BZ is physically significant.
K at zone boundary gives standing wave.

7. Group Velocity
Group velocity:
1-D:
vG = 0 at zone boundaries

8. Derivation of Force Constants from Experiment
If planes up to the pth n.n. interact,
Force on sth plane =
If ωK is known, Cq can be obtained as follows:
Prob 4.4

9. Two Atoms per Primitive Basis→

10. Ka → 0:
Gap
Ka → π:
(M1 >M2 )
Transverse case:
TO branch, Ka → 0:
TA branch, Ka → 0:

11. p atoms in primitive cell → d p branches of dispersion.
d = 3 → 3 acoustical : 1 LA + 2 TA
(3p –3) optical: (p–1) LO + 2(p–1) TO
E.g., Ge or KBr: p = 2 → 1 LA + 2 TA + 1 LO + 2 TO branches Ge
KBr
Number of allowed K in 1st BZ = N

12. Quantization of Elastic Waves
Quantization of harmonic oscillator of angular frequency ω →
Classical standing wave:
Virial theorem: For a power-law potential V ~ xp
For a harmonic oscillator, p = 2,

13. Phonon Momentum Phonon DOFs involve relative coordinates
→ phonons do not carry physical linear momenta ( except for K = G modes )
Reminder: K = G  K = 0 when restricted to 1st BZ .
Proof:
See 7th ed.
Scattering of a phonon with other particles behaves as if it has momentum  K
E.g., elastic scattering of X-ray:
G = reciprocal lattice vector
( whole crystal recoil with momentum  G / Bragg reflection)
Inelastic scattering with a phonon created:
Normal Process: G = 0.
Umklapp Process: G  0.
Inelastic scattering with a phonon absorbed:

14. Inelastic Scattering by Phonons
Neutron scattering:
Conservation of momentum:
Conservation of energy: