the Debye model
/ duh-BYE /
Why does a solid's heat capacity fall toward zero as it approaches absolute zero, in flat contradiction to the classical rule of Dulong and Petit? The atoms in a crystal are not independent oscillators but a coupled network whose collective vibrations are sound waves. The Debye model quantizes these sound waves into phonons and treats a solid almost exactly as one treats blackbody radiation — a Bose gas of quantized waves — nailing the low-temperature heat capacity that Einstein's simpler model got wrong.
Phonons are the quanta of lattice vibration: bosons with mu = 0, since their number is not conserved. Debye makes one bold approximation, replacing the true, complicated dispersion relation with a linear one, omega = v_s k (sound speed times wavevector), for all three acoustic polarizations, and then imposes a maximum frequency omega_D — the Debye frequency — chosen so that the total number of modes equals exactly 3N, the correct number of degrees of freedom for N atoms. Summing the Bose-Einstein energy over these modes gives a heat capacity with two famous limits: at high temperature C -> 3Nk, recovering Dulong-Petit; at low temperature C is proportional to (T/Theta_D)^3, the celebrated Debye T-cubed law, where Theta_D = hbar omega_D/k is the Debye temperature. The T-cubed behavior arises because only long-wavelength acoustic modes are thermally excited near T = 0.
The Debye model quantitatively explains the heat capacity of insulating crystals across the whole temperature range with a single material parameter Theta_D, and its low-temperature T-cubed law is one of the robust triumphs of quantum statistical mechanics. Two honest caveats: the linear-dispersion assumption is only an approximation, and real phonon spectra deviate at short wavelengths — but the T-cubed law survives because it depends only on the long-wavelength acoustic modes, which are always linear. In metals one must add the electronic term (linear in T), so the low-T heat capacity is C = gamma T + A T^3.
Diamond has an unusually high Debye temperature of about 2200 K because its atoms are light and stiffly bonded, so at room temperature it is still well below Theta_D and its heat capacity is far short of the Dulong-Petit value of 3Nk. Lead, with Theta_D about 100 K, has already reached the classical plateau at room temperature.
The Debye temperature sets where a solid crosses from quantum (T-cubed) to classical (Dulong-Petit) behavior.
The Debye model improves on the Einstein model (which assumes all atoms vibrate at ONE frequency) by using the correct linear acoustic dispersion at low frequency; that is exactly what produces the T-cubed law instead of Einstein's exponentially wrong low-T behavior. The linear dispersion is an idealization valid only for long wavelengths.