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. Harmonic approximation: quadratic hamiltonian : elementary excitations.
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Harmonic approximation: quadratic hamiltonian : elementary excitations
Electrons, polarons & excitons are quasi-particles
Entire plane of atoms moving in phase → 1-D problem
Force on sth plane =
(only neighboring planes interact )
Equation of motion:
Propagation along high symmetry directions → 1-D problem
E.g. , , ,  in sc lattice.
Only K 1st BZ is physically significant.
K at zone boundary gives standing wave.
vG = 0 at zone boundaries
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:
Two Atoms per Primitive Basis
Ka → 0:
Ka → π:
(M1 >M2 )
TO branch, Ka → 0:
TA branch, Ka → 0:
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
Number of allowed K in 1st BZ = N
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,
Phonon DOFs involve relative coordinates
→ phonons do not carry physical linear momenta ( except for K = Gmodes )
Reminder: K = G K = 0when restricted to 1st BZ .
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: G0.
Inelastic scattering with a phonon absorbed:
Conservation of momentum:
Conservation of energy: