next up previous
Next: Debye Model of Crystals Up: Solid State Previous: Harmonic Oscillator

Einstein Model of Crystals

Assumptions:

1.
Each atom has a certain equilibrium position in the lattice
2.
Vibrations are harmonic with frequency $\omega$

So we have 3N harmonic oscillators, and heat capacity is

\begin{displaymath}
C =
 3Nk_B\left(\frac{\Theta_E}{T}\right)^2\frac{\exp(-\Theta_E/T)}{\bigl(1
 -\exp(-\Theta_E/T)\bigr)^2} \end{displaymath}

(Change NkB by R to obtain per mole value).
$\mbox{
 \fontsize{12pt}{1}\selectfont
 \InputIfFileExists{einstein.pslatex}{}{}
}$
At $T\gg\Theta_E$ we recover classical behavior (C=3NkB). At low T vibration is frozen: $C\to0$.

Problem With Einstein Model:
At low T heat capacity is $C\propto T^{-2}\exp(-\Theta_E/T)$. This is too fast! The experimental result is $C\propto T^3$.
Reason:
Atoms move cooperatively: if atom A goes to the left, its neighbor might accommodate by going the same way! Einstein model does not account for this.

We must introduce collective motion.



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