In the Debye model, the dispersion relation is linear, ω = c|k|, and the density of states is quadratic as it is in the long wavelength limit.
$$D(\omega )= \frac{3\omega^2}{2\pi^2 c^3}\quad \text{[s rad}^{-1}\text{ m}^{-3}\text{]}.$$Here c is the speed of sound. This holds up to a maximum frequency called the Debye frequency ωD. In three dimensions there are 3 degrees of freedom per atom so the total number of phonon modes is 3n.
$$3n=\int\limits_0^{\omega_D}D(\omega)d\omega.$$Here n is the atomic density. There are no phonon modes with a frequency above the Debye frequency. The Debye freqency is $\omega_D^3 = 6\pi^2nc^3$.
The form below generates a table of where the first column is the angular frequency ω in rad/s and the second column is the density of states D(ω) in units of s/(rad m³).
|
Material | Speed of sound [m/s] | Atom density [m-3] | Debye frequency [rad/sec] |
aluminum | 6320 | 6.03×1028 | 9.66×1013 |
copper | 4660 | 8.47×1028 | 7.98×1013 |
diamond | 18000 | 17.70×1028 | 39.39×1013 |
gold | 3240 | 5.91×1028 | 4.92×1013 |
lead | 2160 | 3.30×1028 | 2.70×1013 |
ice(-4C) | 3280 | 3.07×1028 | 4.00×1013 |
zinc | 4170 | 6.57×1028 | 6.56×1013 |
titanium | 6100 | 5.66×1028 | 9.13×1013 |
tin | 3320 | 3.69×1028 | 4.31×1013 |
silicon | 9620 | 4.99×1028 | 13.81×1013 |
nickel | 5630 | 9.14×1028 | 9.88×1013 |
iron oxide(magnetite) | 5890 | 1.34×1028 | 5.45×1013 |
tungsten | 5180 | 6.42×1028 | 8.08×1013 |
zirconium | 4650 | 4.34×1028 | 6.37×1013 |
molybdenum | 6250 | 6.39×1028 | 9.74×1013 |
beryllium | 12900 | 12.29×1028 | 25.01×1013 |
The density of states can be used to calculate the temperature dependence of thermodynamic quantities.
The energy spectral density is,
$$u(\omega) = \frac{3\omega^2}{2\pi^2 c^3}\frac{\hbar\omega}{\exp\left(\frac{\hbar\omega}{k_BT}\right) -1}.$$In the high temperature limit, the exponential factor can be expanded as $\exp\left(\frac{\hbar\omega}{k_BT}\right)\approx 1 + \frac{\hbar\omega}{k_BT}$. The energy spectral density then becomes,
$$u(\omega) = \frac{3\omega^2}{2\pi^2 c^3}k_BT.$$This can be integrated to yield the internal energy density,
$$u = \frac{\omega_D^3}{2\pi^2 c^3}k_BT= 3nk_BT.$$The specific heat has the Dulong-Petit form,
$$c_v = 3nk_B.$$