Stat Mech 4 - Lattice Heat Capacity of Solids

Dulong-Petit Law

States that the molar heat capacity of most monatomic solids is approximately 3R per mole, or 3k_B per tom at high temperatures.

The Einstein model of solids

Theoretical model that describes the heat capacity of solids by considering quantized harmonic oscillators, predicting that the heat capacity approaches a constant value at high temperatures.

Assume that atoms are weakly interacting and are all in their own mean position, with no dynamic interaction between them.

Einstein Model at high temperatures

recover the Dulong-Petit Law - c_v ~ 3Nk

Einstein model at low temperatures

As T approaches 0, c_v decreases exponentially, faster than seen experimentally, so the model fails at low temperatures.

The Debye Model

Built on Einstein’s model, as atoms in the lattice as independent oscillators, but recognises the weakly interacting nature of the bonded atoms is required for the form of the collective wave phenomena in the crystal.

Think of linear chain of N atoms with separation a connected by springs and apply periodic boundary conditions.

Debye model at high temperatures

recovers Dulong-Petit Law, c_v ~ 3Nk

Debye model at low temperatures

recovers that c_v is proportional to T³