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

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

(Change Nk_B by R to obtain per mole value).

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At $T\gg\Theta_E$ we recover classical behavior (C=3Nk_B). 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.