Debye Specific Heat Theory

Introduction

The Debye model describes the vibrational modes in a solid by treating them as acoustic phonons. Unlike the Einstein model, which assumes a single frequency for all oscillators, the Debye model considers a continuous spectrum of vibrational frequencies.

Phonon Density of States

The phonon density of states \( g(\omega) \) is proportional to \( \omega^2 \) up to a maximum frequency \( \omega_D \):

\( g(\omega) = \frac{V \omega^2}{2 \pi^2 \left( \frac{\hbar}{v} \right)^3} \)

where \( V \) is the volume, \( \hbar \) is Planck's constant, and \( v \) is the velocity of sound.

Debye Temperature

The Debye temperature \( \theta_D \) is given by:

\( \theta_D = \frac{\hbar \omega_D}{k_B} \)

where \( k_B \) is Boltzmann's constant.

Heat Capacity Calculation

The heat capacity at constant volume \( C_V \) is given by:

\( C_V = \int_0^{\omega_D} \frac{\partial \langle E \rangle}{\partial T} g(\omega) \, d\omega \)

where \( \langle E \rangle \) is the average energy of a phonon mode:

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

Low-Temperature Limit

At low temperatures, \( T \ll \theta_D \), the heat capacity \( C_V \) follows a \( T^3 \) dependence:

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

where \( N \) is the number of atoms.

High-Temperature Limit

At high temperatures, \( T \gg \theta_D \), the heat capacity approaches:

\( C_V \approx 3Nk_B \)