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Harmonic Oscillator

Phonons

$\mbox{ \InputIfFileExists{oscillator.pstex_t}{}{} }$

Classical description: energy might have any value. One degree of freedom--average energy k_BT.

Quantum description: only portions of $\hbar\omega$ , with $\omega=\sqrt{k/m}$ are allowed! Energy:

$\begin{displaymath} E = \hbar\omega\left(n+\frac12\right),\qquad n=0,1,2,\dots\end{displaymath}$

We say that the system has n phonons.

System might have as many phonons, as it wants (Bose-Einstein statistics).

Thermodynamics

Partition Function

$\begin{displaymath} \begin{gathered} Q = \exp(-\beta\hbar\omega/2) + \exp(-3\be... ...ega/2) +\ \exp(-5\beta\hbar\omega/2) + \ldots \end{gathered}\end{displaymath}$

Geometric progression:

$\begin{displaymath} a+ab+ab^2+ab^3+\dots = \frac{a}{1-b}\end{displaymath}$

We have:

$\begin{displaymath} Q = \frac{\exp(-\beta\hbar\omega/2)}{1-\exp(-\beta\hbar\omega)}\end{displaymath}$

Einstein Temperature:: Let $\Theta_E = \hbar\omega/k_B$ . Then
$\begin{displaymath} Q = \frac{\exp(-\Theta_E/2T)}{1-\exp(-\Theta_E/T)} \end{displaymath}$
$\Theta_E$ is Einstein temperature.

Thermodynamic Quantities

Free energy:

$\begin{displaymath} A = \frac{k_B\Theta_E}{2} + k_BT\ln\bigl(1-\exp(-\Theta_E/T)\bigr)\end{displaymath}$

Entropy:

$\begin{displaymath} S = - k_B\ln\bigl(1-\exp(-\Theta_E/T)\bigr) + \frac{k_B\Theta_E\exp(-\Theta_E/T)}{T\bigl(1-\exp(-\Theta_E/T)\bigr)} \end{displaymath}$

Energy:

$\begin{displaymath} E=A+TS = \frac{k_B\Theta_E}{2} + \frac{k_B\Theta_E\exp(-\Theta_E/T)}{1-\exp(-\Theta_E/T)} \end{displaymath}$

Heat capacity:

$\begin{displaymath} C = k_B\left(\frac{\Theta_E}{T}\right)^2\frac{\exp(-\Theta_E/T)}{\bigl(1 -\exp(-\Theta_E/T)\bigr)^2} \end{displaymath}$

Asymptotics

High Temperatures

$\begin{displaymath} T\gg\Theta_E\end{displaymath}$

we obtain

$\begin{displaymath} \exp(-\Theta_E/T)\approx1-\frac{\Theta_E}{T}\end{displaymath}$

Then energy and heat capacity are

$\begin{displaymath} E\approx\frac{k_B\Theta_E}{2} + k_BT,\quad C\approx k_B\end{displaymath}$

This is exactly classical result!

Low Temperatures

$\begin{displaymath} T\ll\Theta_E\end{displaymath}$

we obtain

$\begin{displaymath} \exp(-\Theta_E/T)\to0\end{displaymath}$

Energy and heat capacity:

$\begin{displaymath} \begin{gathered} E\approx \frac{k_B\Theta_E}{2} + k_B\Thet... ...ft(\frac{\Theta_E}{T}\right)^2\exp(-\Theta_E/T) \end{gathered}\end{displaymath}$

The vibrations are ``frozen'' at low temperatures!