Derivation of Einstein's Specific Heat Formula for Solids

Below is a step-by-step derivation of Einstein’s specific heat formula for a solid, modeling atoms as harmonic oscillators vibrating at a single frequency, followed by high and low temperature cases.

Step 1: Model Assumptions

Einstein’s model assumes a solid with \( N \) atoms, each acting as a 3D harmonic oscillator, giving \( 3N \) oscillators, all vibrating at frequency \( \omega \). Each oscillator is quantized.

Step 2: Energy of a Single Oscillator

The energy levels of a quantum harmonic oscillator are:

\[ E_n = \left( n + \frac{1}{2} \right) \hbar \omega, \quad n = 0, 1, 2, \dots \]

The average thermal energy (excluding zero-point energy) is:

\[ \langle E \rangle_{\text{thermal}} = \frac{\hbar \omega}{e^{\hbar \omega / k_B T} - 1} \]

Step 3: Total Internal Energy \( U \)

For \( 3N \) oscillators:

\[ U = 3N \langle E \rangle_{\text{thermal}} = 3N \frac{\hbar \omega}{e^{\hbar \omega / k_B T} - 1} \]

Step 4: Specific Heat \( C_V \)

The specific heat at constant volume is:

\[ C_V = \left( \frac{\partial U}{\partial T} \right)_V \]

Let \( \beta = \frac{\hbar \omega}{k_B T} \). Then:

\[ U = 3N \frac{\hbar \omega}{e^{\beta} - 1} \]

Differentiate with respect to \( T \), noting \( \frac{\partial \beta}{\partial T} = -\frac{\beta}{T} \):

\[ C_V = 3N \hbar \omega \cdot \frac{e^{\beta} \cdot \frac{\beta}{T}}{(e^{\beta} - 1)^2}\] \[ C_V = 3N k_B \left( \frac{\hbar \omega}{k_B T} \right)^2 \frac{e^{\hbar \omega / k_B T}}{(e^{\hbar \omega / k_B T} - 1)^2} \]

Define \( \theta_E = \frac{\hbar \omega}{k_B} \), so:

\[ C_V = 3N k_B \left( \frac{\theta_E}{T} \right)^2 \frac{e^{\theta_E / T}}{(e^{\theta_E / T} - 1)^2} \]

This is the Einstein specific heat formula.

Case 1: High Temperature (\( T \gg \theta_E \))

When \( \frac{\theta_E}{T} \ll 1 \), \( e^{\theta_E / T} \approx 1 + \frac{\theta_E}{T} \), so:

\[ \frac{e^{\theta_E / T}}{(e^{\theta_E / T} - 1)^2} \approx \frac{1 + \frac{\theta_E}{T}}{\left( \frac{\theta_E}{T} \right)^2} \approx \frac{\theta_E}{T} \]

Thus:

\[ C_V \approx 3N k_B \left( \frac{\theta_E}{T} \right)^2 \cdot \frac{\theta_E}{T} = 3N k_B \]

This is the Dulong-Petit limit, \( 3R \) per mole.

Case 2: Low Temperature (\( T \ll \theta_E \))

When \( \frac{\theta_E}{T} \gg 1 \), \( e^{\theta_E / T} \gg 1 \), so:

\[ C_V \approx 3N k_B \left( \frac{\theta_E}{T} \right)^2 e^{-\theta_E / T} \]

The specific heat decreases exponentially, which overestimates the drop compared to the experimental \( T^3 \) behavior.

Explanation

The derivation assumes \( 3N \) harmonic oscillators vibrating at a single frequency, computes the quantum mechanical average energy, and derives the specific heat. The high-temperature case yields the classical limit, while the low-temperature case shows an exponential decay, a limitation improved by the Debye model.

Related Topics:

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