ME 381R Lecture 5: Phonons Dr. Li Shi Department of Mechanical Engineering The University of Texas at Austin Austin, TX 78712 www.me.utexas.edu/~lishi lishi@mail.utexas.edu

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:

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

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

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