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Subsections


Harmonic Oscillator

Phonons

$\mbox{
 \InputIfFileExists{oscillator.pstex_t}{}{}
}$
Classical description: energy might have any value. One degree of freedom--average energy kBT.

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

If

\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

If

\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!



© 1997 Boris Veytsman and Michael Kotelyanskii
Wed Oct 1 00:45:35 EDT 1997