At temperatures well below the Debye temperature, how does the heat capacity of an insulating crystal (the lattice heat capacity) change with temperature?

Answer: B. It decreases in proportion to the cube of the temperature

The energy of lattice vibrations is quantized, and vibrations whose energy unit ħω is larger than kT are hardly excited. At temperatures well below the Debye temperature, only low-energy, long-wavelength acoustic phonons are excited, and their number grows with temperature, so the lattice heat capacity is proportional to T³ (the Debye T³ law). A nearly constant 3R is the classical limit at high temperatures comparable to or above the Debye temperature, corresponding to the Dulong–Petit law. A heat capacity proportional to temperature is the signature of the electronic heat capacity from conduction electrons in metals, which arises because only electrons near the Fermi level are thermally excited.