Debye model

Phonon continuum approximation with a Debye cutoff; produces the low-temperature heat capacity law and the high-temperature Dulong–Petit limit.

Debye model

Physical setup

The Debye model treats lattice vibrations as a continuum of quantized normal modes (phonons) with approximately linear dispersion $\omega\approx v_s|k|$ at small $|k|$ in three dimensions. Instead of $3N$ identical modes as in the Einstein solid , one uses a distribution of frequencies up to a cutoff $\omega_D$ chosen so that the total number of modes is $3N$ .

Thermodynamic quantities are computed in the canonical ensemble at temperature $T$ .

Debye density of states

In the isotropic 3D Debye approximation, the vibrational density of states is taken as

g(\omega)=\frac{9N}{\omega_D^3}\,\omega^2,\qquad 0\le \omega\le \omega_D,

and $g(\omega)=0$ for $\omega>\omega_D$ .

Define the Debye temperature

\Theta_D=\frac{\hbar\omega_D}{k_B}.

Internal energy and free energy

For each mode of frequency $\omega$ , the mean energy is that of a quantum harmonic oscillator :

u_\omega(T)=\frac{\hbar\omega}{2}+\frac{\hbar\omega}{e^{\beta\hbar\omega}-1}.

Thus

U(T)=\int_0^{\omega_D} g(\omega)\,u_\omega(T)\,d\omega.

The zero-point contribution $\frac12\int_0^{\omega_D} g(\omega)\hbar\omega\,d\omega$ is independent of $T$ and does not affect $C_V$ .

A convenient free-energy representation (up to an additive zero-point constant) is

F(T)=k_B T \int_0^{\omega_D} g(\omega)\,\ln\!\bigl(1-e^{-\beta\hbar\omega}\bigr)\,d\omega.

Heat capacity formula

Differentiating $U(T)$ gives the Debye heat capacity (see heat capacity at constant volume ):

C_V(T)=9N k_B\left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D/T} \frac{x^4 e^x}{(e^x-1)^2}\,dx, \qquad x=\beta\hbar\omega.

Low- and high-temperature limits

Low temperature $T\ll \Theta_D$ (upper limit $\Theta_D/T\to\infty$ ):
$\int_0^\infty \frac{x^4 e^x}{(e^x-1)^2}\,dx=\frac{4\pi^4}{15},$
hence
$C_V(T)\sim \frac{12\pi^4}{5}N k_B\left(\frac{T}{\Theta_D}\right)^3,$
giving the $T^3$ law.
High temperature $T\gg \Theta_D$ :
$C_V(T)\to 3N k_B,$
recovering the Dulong–Petit limit (also captured by Einstein solid ).

Conceptual takeaway

Relative to the Einstein model, the key change is the abundance of low-frequency modes (the $\omega^2$ density of states), which produces a power-law rather than exponential suppression of thermal excitations at low $T$ .