Categories: Thermodynamics | Solid state physics

Debye model

In thermodynamics and solid state physics, the Debye model is a method of calculating the phonon contribution to the specific heat in a solid. It improves upon the Einstein model, which assumes a single phonon frequency, by approximating the phonon density of states as a constant up to a cutoff frequency, called the Debye frequency. This model correctly predicts the low temperature dependence of the heat capacity, which is proportional to T³, and it also recovers the Dulong-Petit law at high temperatures.

According to the Debye model, which was developed by Peter Debye in 1913,

$C_V(T)=\begin{cases} \frac{12}{5}\pi^4k_BN(\frac{T}{\theta_D})^3, & \mbox{if }T<<\theta_D \\ 3Nk_B, & \mbox{if }T>>\theta_D \end{cases}$

where $θ D$ is the Debye temperature, which is characteristic for each material. The following table lists Debye temperatures for several metals:

Substance	$θ D (K)$
Aluminum	426
Cadmium	186
Chromium	610
Copper	344.5
Gold	165
$α$ -Iron	464
Lead	96
$α$ -Manganese	476
Nickel	440
Platinum	240
Silicon	640
Silver	225
Tin (white)	195
Titanium	420
Tungsten	405
Zinc	300
Diamond	2200
Ice	192

References

CRC Handbook of Chemistry and Physics, 56th Edition (1975-1976)

Categories: Thermodynamics | Solid state physics