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## PHY452H1S Basic Statistical Mechanics. Lecture 21: Phonon modes. Taught by Prof. Arun Paramekanti

Posted by peeterjoot on April 4, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

# Disclaimer

Peeter’s lecture notes from class. May not be entirely coherent.

These are notes for the last class, which included a lot of discussion not captured by this short set of notes (as well as slides which were not transcribed).

# Phonon modes

If we model a solid as a set of interconnected springs, as in fig. 1.1, then the potentials are of the form

Fig 1.1: Solid oscillator model

\begin{aligned}V = \frac{1}{{2}} C \sum_n \left( {u_n - u_{n+1}} \right)^2,\end{aligned} \hspace{\stretch{1}}(1.2.1)

with kinetic energies

\begin{aligned}K = \sum_n \frac{p_n^2}{2m}.\end{aligned} \hspace{\stretch{1}}(1.2.2)

It’s possible to introduce generalized forces

\begin{aligned}F = -\frac{\partial {V}}{\partial {u_n}}\end{aligned} \hspace{\stretch{1}}(1.2.3)

Can differentiate

\begin{aligned}m \frac{d^2 u_n}{dt^2} = - C \left( { u_n - u_{n+1}} \right)- C \left( { u_n - u_{n-1}} \right)\end{aligned} \hspace{\stretch{1}}(1.2.4)

Assuming a Fourier representation

\begin{aligned}u_n = \sum_k \tilde{u}(k) e^{i k n a},\end{aligned} \hspace{\stretch{1}}(1.2.5)

we find

\begin{aligned}m \frac{d^2 \tilde{u}(k)}{dt^2} = - 2 C \left( { 1 - \cos k a} \right)\tilde{u}(k)\end{aligned} \hspace{\stretch{1}}(1.2.6)

This looks like a harmonic oscillator with

\begin{aligned}\omega(k) = \sqrt{ \frac{2 C}{m} ( 1 - \cos k a)}.\end{aligned} \hspace{\stretch{1}}(1.2.7)

This is plotted in fig. 1.2. In particular note that for for $k a \ll 1$ we can use a linear approximation

Fig 1.2: Angular frequency of solid oscillator model

\begin{aligned}\omega(k) \approx \sqrt{ \frac{C}{m} a^2 } \left\lvert {k} \right\rvert.\end{aligned} \hspace{\stretch{1}}(1.2.8)

Experimentally, looking at specific for a complex atomic structure like Gold, we find for example good fit for a model such as

\begin{aligned}C \sim \underbrace{A T}_{\text{Contribution due to electrons.}}+ \underbrace{B T^3}_{\text{Contribution due to phonon like modes where there are linear energy momenum relations.}}.\end{aligned} \hspace{\stretch{1}}(1.2.9)