Sketch the specific heat curve at constant volume, Cv, as a...

Sketch the specific heat curve at constant volume, $C_v$, as a function of the absolute temperature, $T$, for a metallic solid. Give an argument showing why the contribution to $C_v$ from the free electrons is proportional to $T$.

Key Concepts

Specific Heat

This concept refers to the amount of heat required to raise the temperature of a material by a given amount. In metallic solids, the total specific heat at constant volume has contributions from both the lattice (phonons) and the free electrons. The lattice contribution typically follows a T³ dependence at low temperatures (as explained by the Debye model) and approaches a constant (the Dulong-Petit limit) at high temperatures.

Debye Model

The Debye model explains the behavior of lattice (phonon) vibrations in a solid and predicts that the heat capacity due to these vibrations varies as T³ at low temperatures. This model helps illustrate how the phonon contribution to the specific heat increases with temperature until it saturates at higher temperatures.

Sommerfeld Theory

Sommerfeld theory describes the electronic contribution to the specific heat in metals. According to this theory, only electrons near the Fermi surface can be thermally excited because of the Pauli exclusion principle. As a result, the number of electrons that participate in thermal excitation is proportional to T, making the electronic contribution to the heat capacity linear in temperature at low temperatures.

Fermi-Dirac Statistics

These statistics govern the occupancy of electron energy states in a metal at finite temperatures. Due to the sharp distribution of electron energies around the Fermi level at low temperatures, only a small fraction of electrons (within an energy range of about kBT) can be excited. This is the fundamental reason behind the T-dependence of the electronic contribution to the specific heat.

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