Derivation of Debye’s Specific Heat Formula for Solids

Below is a step-by-step derivation of the Debye specific heat formula for a solid, based on the Debye model of lattice vibrations, followed by analysis for low and high temperature cases.

Step 1: Phonon Density of States in Debye Model

The Debye model treats the solid as an isotropic elastic continuum with a linear dispersion relation \(\omega = v k\), where \(\omega\) is the frequency, \(v\) is the average speed of sound, and \(k\) is the wave vector magnitude.

The number of phonon modes in a volume \(V\) for three polarizations (one longitudinal, two transverse) with wave vectors between \(k\) and \(k + dk\) is:

\[\frac{3V}{(2\pi)^3} 4\pi k^2 dk\]

Substitute \(k = \omega / v\), \(dk = d\omega / v\):

\[g(\omega) d\omega = \frac{3V \omega^2 d\omega}{2\pi^2 v^3}\]

where \(g(\omega)\) is the density of states.

The total number of modes equals \(3N\) (for \(N\) atoms):

\[\int_0^{\omega_D} g(\omega) d\omega = 3N\]

Integrate:

\[\frac{3V}{2\pi^2 v^3} \int_0^{\omega_D} \omega^2 d\omega = \frac{3V}{2\pi^2 v^3} \cdot \frac{\omega_D^3}{3} = \frac{V \omega_D^3}{2\pi^2 v^3} = 3N\]

Thus:

\[\omega_D^3 = \frac{6\pi^2 v^3 N}{V}\]

Rewrite the density of states:

\[g(\omega) = \frac{9N \omega^2}{\omega_D^3}\]

for \(\omega \leq \omega_D\).

Step 2: Total Internal Energy \(U\)

Each phonon mode behaves as a boson with average energy (ignoring zero-point energy):

\[\langle E(\omega) \rangle = \frac{\hbar \omega}{e^{\hbar \omega / k_B T} - 1}\]

The total vibrational energy \(U\) is:

\[U = \int_0^{\omega_D} g(\omega) \langle E(\omega) \rangle d\omega\] \[U= \int_0^{\omega_D} \frac{9N \omega^2}{\omega_D^3} \cdot \frac{\hbar \omega}{e^{\hbar \omega / k_B T} - 1} d\omega\]

Simplify:

\[U = \frac{9N \hbar}{\omega_D^3} \int_0^{\omega_D} \frac{\omega^3 d\omega}{e^{\hbar \omega / k_B T} - 1}\]

Step 3: Change of Variable to Dimensionless Form

Define \(x = \frac{\hbar \omega}{k_B T}\), so \(\omega = \frac{k_B T}{\hbar} x\), \(d\omega = \frac{k_B T}{\hbar} dx\). The limits change from \(\omega = 0\) to \(\omega = \omega_D\) to \(x = 0\) to \(x = x_D = \frac{\hbar \omega_D}{k_B T} = \frac{\theta_D}{T}\), where \(\theta_D = \frac{\hbar \omega_D}{k_B}\).

Substitute:

\[\omega^3 d\omega = \left( \frac{k_B T}{\hbar} x \right)^3 \cdot \frac{k_B T}{\hbar} dx = \left( \frac{k_B T}{\hbar} \right)^4 x^3 dx\]

The integral becomes:

\[\int_0^{\omega_D} \frac{\omega^3 d\omega}{e^{\hbar \omega / k_B T} - 1} = \left( \frac{k_B T}{\hbar} \right)^4 \int_0^{x_D} \frac{x^3 dx}{e^x - 1}\]

Thus:

\[U = \frac{9N \hbar}{\omega_D^3} \cdot \left( \frac{k_B T}{\hbar} \right)^4 \int_0^{x_D} \frac{x^3 dx}{e^x - 1}\] \[U= 9N k_B \left( \frac{T}{\theta_D} \right)^3 T \int_0^{x_D} \frac{x^3 dx}{e^x - 1}\]

Step 4: Specific Heat \(C_V = \frac{\partial U}{\partial T}\)

The specific heat at constant volume is the derivative of \(U\) with respect to \(T\). For each mode, the contribution to specific heat is:

\[k_B \left( \frac{\hbar \omega}{k_B T} \right)^2 \frac{e^{\hbar \omega / k_B T}}{(e^{\hbar \omega / k_B T} - 1)^2}\] \[= k_B \frac{x^2 e^x}{(e^x - 1)^2}\]

Integrating over the density of states:

\[C_V = \int_0^{\omega_D} g(\omega) k_B \frac{x^2 e^x}{(e^x - 1)^2} d\omega\]

Substitute \(g(\omega) = \frac{9N \omega^2}{\omega_D^3}\), and using \(\omega^2 d\omega = \left( \frac{k_B T}{\hbar} \right)^3 x^2 dx\):

\[C_V = \frac{9N}{\omega_D^3} \left( \frac{k_B T}{\hbar} \right)^3 k_B \int_0^{x_D} x^2 \cdot \frac{x^2 e^x}{(e^x - 1)^2} dx \] \[C_V = 9N k_B \left( \frac{T}{\theta_D} \right)^3 \int_0^{x_D} \frac{x^4 e^x}{(e^x - 1)^2} dx\]

This is the Debye specific heat formula.

Case 1: Low Temperature (\( T \ll \theta_D \))

At low temperatures, \( T \ll \theta_D \), so \( x_D = \frac{\theta_D}{T} \to \infty \). The upper limit of the integral in the specific heat expression can be approximated as infinity:

\[C_V \approx 9N k_B \left( \frac{T}{\theta_D} \right)^3 \int_0^\infty \frac{x^4 e^x}{(e^x - 1)^2} dx\]

The integral is a known result in the Debye model:

\[\int_0^\infty \frac{x^4 e^x}{(e^x - 1)^2} dx = \frac{4\pi^4}{15}\]

Thus:

\[C_V \approx 9N k_B \left( \frac{T}{\theta_D} \right)^3 \cdot \frac{4\pi^4}{15}\] \[C_V \approx \frac{12\pi^4 N k_B}{5} \left( \frac{T}{\theta_D} \right)^3\]

This gives the characteristic \( C_V \propto T^3 \) behavior, which matches experimental observations for the specific heat of solids at low temperatures.

Case 2: High Temperature (\( T \gg \theta_D \))

At high temperatures, \( T \gg \theta_D \), so \( x_D = \frac{\theta_D}{T} \to 0 \). For small \( x \), the integrand can be approximated:

\[\frac{x^4 e^x}{(e^x - 1)^2} \approx \frac{x^4 \cdot (1 + x + \cdots)}{(x + \cdots)^2} \approx \frac{x^4}{x^2} = x^2\]

The integral becomes:

\[\int_0^{x_D} \frac{x^4 e^x}{(e^x - 1)^2} dx \approx \int_0^{x_D} x^2 dx = \frac{x_D^3}{3}\] \[= \frac{1}{3} \left( \frac{\theta_D}{T} \right)^3\]

Thus:

\[C_V \approx 9N k_B \left( \frac{T}{\theta_D} \right)^3 \cdot \frac{1}{3} \left( \frac{\theta_D}{T} \right)^3\] \[= 9N k_B \cdot \frac{1}{3} = 3N k_B\]

This is the classical Dulong-Petit limit, where the specific heat per mole approaches \( 3R \) (where \( R = N_A k_B \)), consistent with the classical equipartition theorem.

Explanation

The derivation starts with the phonon density of states, enforces the mode cutoff via the Debye frequency, computes the total energy using Bose-Einstein statistics, changes to dimensionless variables for simplification, and differentiates to obtain \( C_V \). The low-temperature case yields the \( T^3 \) dependence, while the high-temperature case recovers the classical limit, demonstrating the Debye model's ability to bridge quantum and classical regimes.

Related Topics:

• Derivation of Classical Specific Heat Formula for Solids
• Derivation of Einsteins Specific Heat Formula for Solids
• Comparison of Einstein and Debye Models for Specific Heat