Debye Model

Debye approximated the normal vibrations with the elastic vibrations of an isotropic continuous body where the number of vibrational modes with frequencies between $\omega$ and $\omega + d\omega$ is

$$g(\omega)=\begin{cases}\xi \omega^2, \hspace{.2cm} \text{when} \h... ...e{.2cm} \omega < \omega_0, \\ 0, \hspace{.2cm} \text{otherwise}, \end{cases}$$ (191)

where $\xi \equiv 3 V/(2 \pi^2 c^3)$ and

$$\int_0^{\omega_0} d \omega g(\omega) = 3 N = \frac{V \omega_0^3}{2\pi^2 C^3}.$$ (192)

Therefore,

$$\xi = 9 N 2\pi^2 c^3/(\omega_0^3 2 \pi^2 c^3).$$ (193)

According to Eqs.(192) and (182),

$$Z = -\int_0^{\omega_0}d\omega \xi \omega^2 \Bigg ( e^{\beta \hbar \omega/2}-e^{-\beta \hbar \omega/2} \Bigg ).$$ (194)

Therefore,

$$Z = -\int_0^{\omega_0}d\omega \xi \omega^2 e^{\beta \hbar \omega/2} -\int_0^{\omega_0} d\omega \xi \omega^2 (1- e^{\beta \hbar \omega} ),$$ (195)

and

$$Z = -\frac{\beta \hbar}{2} \xi \frac{\omega_0^4}{4}-\int_0^{\omega_0} d\omega \xi \omega^2 (1- e^{\beta \hbar \omega} ).$$ (196)

The internal energy $E$ is computed from Eqs. (196) and (193) as follows,

$$E = \frac{\partial \text{ln} Z}{\partial (-\beta)} = \frac{\hbar}... ... \xi \omega^3 \hbar \frac{e^{-\beta \hbar \omega}}{1- e^{-\beta \hbar \omega}},$$ (197)

and introducing the change of variables $\gamma \equiv \beta \hbar \omega$,

$$E = \frac{\hbar}{2} \frac{9 N \omega_0}{4} + \frac{1}{\beta \hbar... ... d \gamma \xi \frac{\gamma^3}{(\beta \hbar)^3} \hbar \frac{1}{(e^{\gamma} -1)}.$$ (198)

Considering that

$$f(x)=\frac{3}{x^3} \int_0^x d\gamma \frac{\gamma^3}{e^{\gamma}-1}... ... 1-\frac{3}{8}x+... & x << 1 \\ \frac{\pi}{5 x^3}+... & x >> 1, \end{cases}$$ (199)

we obtain, according to Eqs. (198) and (199),

$$\tilde{E} = E - \frac{\hbar}{2} \frac{9 N \omega_0}{4} = \frac{\o... ...5 (\hbar \omega_0)^3} & \text{when} \hspace{.1cm} T \rightarrow 0. \end{cases}$$ (200)

Therefore the Debye model predicts the following limits for the heat capacity of a solid lattice,

$$C_v= \begin{cases} 3 N k & \text{when} \hspace{.1cm} T \rightar... ...hbar \omega_0)^3} T^3 & \text{when} \hspace{.1cm} T \rightarrow 0. \end{cases}$$ (201)

which are the correct high and low temperature limits, represented by the following diagram: