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Einstein Model of Crystals


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

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

C =
 -\exp(-\Theta_E/T)\bigr)^2} \end{displaymath}

(Change NkB by R to obtain per mole value).
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 $C\propto T^3$.
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