1. ME 381R Lecture 5: Phonons Dr. Li Shi
Department of Mechanical Engineering
The University of Texas at Austin
Austin, TX 78712

2. Classical vs. Quantum Oscillator
Classical Harmonic Oscillator
Allowed energy states
n = 0, 1, 2,…
Quantum Oscillator
The state of the particle is associated with a wave function , whose modulus squared |(x)|2 gives the probability of finding the particle at x
Energy is quantized, and ħw is
a quntum of energy
Schrodinger equation: m
Frictionless
The state of a particle at time t is specified by location x(t) and momentum p(t)
Newton’s 2nd law:
Displacement:
: amplitude
Energy:

3. Crystal Vibration & Phonons
Thermal vibrations in crystals are thermally exited phonons, like the thermally excited photons of black-body electromagnetic radiation in a cavity
The energy of a crystal vibration mode at
frequency w is
when the mode is exited to quantum number n (the mode is occupied by n phonons).
A phonon is a quantum (ħw) of crystal vibration energy, as an analogue to photon
n follows the Bose-Einstein statistics:
Equilibrium distribution

4. Total Energy of Crystal Vibration
p: polarization(LA,TA, LO, TO)
K: wave vector

5. Phonon Momentum The physical momentum of a crystal is: p = M(d/dt)us
It can be shown that if the crystal carriers a phonon K, its physical momentum is 0 unless K =0.
For practical purpose, a phonon acts as if its momentum were ћK. For example, a photon or a neutron incident on a crystal can generate a phonon that meets with the selection rule
k’ + K = k + G
Alternatively, the photon or neutron can also absorb a phonon
k’ = k + K + G
These two processes are inelastic (Raman) scattering processes of photons, as opposed to the elastic scattering process (K=0)
Incident photon
Scattered photon
Reciprocal lattice vector
Phonon