PHZ3400 Phonons

Law of Dulong and Pettit: failure of classical physics

$C_{v}={\frac {\partial \epsilon (T)}{\partial T}}=3nk_{B}$

-Works fine at high temperature, but at low temperature the quantum mechanical nature of the material dominates. The Debye model gives the correct interpretation at low temperature.

Quantization of sound waves: phonons

Bose-Einstein distribution

$n(\mathrm {B} \hbar \omega )={\frac {1}{e^{\mathrm {B} \hbar \omega }-1}}$

Thermodynamics of phonons

Debye model

- Analogous to Planck's Blackbody Radiation Law, except this deals with phonons instead of photons.

Planck distribution and the black-body radiation law

Following Kittel's derivation, consider a set of harmonic oscillators expressed as a ratio between (n) and (n+1) quantum states. ${\frac {N_{1+n}}{N_{n}}}=e^{-{\frac {\hbar \omega }{\tau }}}$ Such that $\tau =K_{B}T$

The fraction of oscillators in the nth quantum states then follows as ${\frac {e^{-{\frac {\hbar \omega }{\tau }}}}{\sum _{s=0}^{\infty }e^{-{\frac {s*\hbar *\omega }{\tau }}}}}$

The average excitation number of the oscillator has the form $<n>=\sum _{s=0}^{\infty }{\frac {e^{-{\frac {s*\hbar *\omega }{\tau }}}s}{\sum _{s=0}^{\infty }e^{-{\frac {s*\hbar *\omega }{\tau }}}}}$

After simplifying summations, this can be rewritten in the form yielding the Planck distribution: $<n>={\frac {1}{e^{-{\frac {\hbar \omega }{\tau }}}-1}}$