Peeter Joot's (OLD) Blog.

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Posts Tagged ‘phonon modes’

A final pre-exam update of my notes compilation for ‘PHY452H1S Basic Statistical Mechanics’, Taught by Prof. Arun Paramekanti

Posted by peeterjoot on April 22, 2013

Here’s my third update of my notes compilation for this course, including all of the following:

April 21, 2013 Fermi function expansion for thermodynamic quantities

April 20, 2013 Relativistic Fermi Gas

April 10, 2013 Non integral binomial coefficient

April 10, 2013 energy distribution around mean energy

April 09, 2013 Velocity volume element to momentum volume element

April 04, 2013 Phonon modes

April 03, 2013 BEC and phonons

April 03, 2013 Max entropy, fugacity, and Fermi gas

April 02, 2013 Bosons

April 02, 2013 Relativisitic density of states

March 28, 2013 Bosons

plus everything detailed in the description of my previous update and before.

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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)

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