Chapter 4: Thermal Properties of Solids

1. Introduction: Phonons and Thermal Properties

In previous chapters, we learned that phonons are quantized lattice vibrations. But why should we care about these quantum mechanical objects? The answer lies in their profound influence on the macroscopic thermal properties of materials. Every time you heat a metal, observe thermal expansion, or measure thermal conductivity, you're witnessing the collective behavior of trillions of phonons.

This chapter bridges the gap between microscopic phonon physics and macroscopic thermal phenomena. We'll see how the distribution of phonon modes—characterized by the density of states we studied in Chapter 3— directly determines properties like heat capacity and thermal expansion.

2. Classical Theory: The Dulong-Petit Law

2.1 Historical Context

In 1819, French scientists Pierre Louis Dulong and Alexis Thérèse Petit made an empirical observation: at high temperatures, the molar heat capacity of many solid elements is approximately constant, around 25 J/(mol·K) or 3R, where R is the gas constant (8.314 J/(mol·K)).

2.2 Classical Derivation

From classical statistical mechanics, we can derive this result. Consider N atoms in a solid, each able to vibrate in three dimensions. Each degree of freedom has average energy k_BT/2 (kinetic) + k_BT/2 (potential) = k_BT by the equipartition theorem.

Total energy for N atoms with 3N degrees of freedom:

The heat capacity at constant volume is:

where we used N = N_A (Avogadro's number) for one mole.

2.3 Limitations of the Classical Theory

However, the Dulong-Petit law fails dramatically in several important cases:

• Low Temperatures: Heat capacity drops to zero as T → 0, not remaining at 3R
• Light Elements: Diamond, beryllium, and silicon have C_V ≪ 3R even at room temperature
• Temperature Dependence: The classical theory predicts no temperature dependence, but experiments show C_V(T) varies strongly

These failures pointed to a fundamental problem: classical physics cannot explain the thermal properties of solids. A quantum theory was needed.

3. Einstein Model of Heat Capacity

3.1 Einstein's Quantum Hypothesis (1907)

Albert Einstein made a revolutionary proposal: treat each atom as an independent quantum harmonic oscillator with a single characteristic frequency ω_E. Unlike classical oscillators that can have any energy, quantum oscillators have discrete energy levels:

The key quantum feature: oscillators at low temperature tend to stay in low energy states.

3.2 Derivation of Einstein Heat Capacity

Using Bose-Einstein statistics, the average number of phonons at temperature T is:

The average energy per oscillator (excluding zero-point energy):

For N atoms (3N oscillators), total energy:

Heat capacity:

Defining the Einstein temperature Θ_E = ℏω_E/k_B:

3.3 Behavior of Einstein Model

High Temperature Limit (T ≫ Θ_E):

When T ≫ Θ_E, e^Θ_E/T ≈ 1 + Θ_E/T, giving:

✓ Recovers the classical Dulong-Petit law!

Low Temperature Limit (T ≪ Θ_E):

✓ Goes to zero as T → 0 (satisfies third law of thermodynamics)

✗ But decreases exponentially, not as T³ as observed experimentally

4. Debye Model of Heat Capacity

4.1 Debye's Improvement (1912)

Peter Debye realized that Einstein's assumption of a single frequency was too simplistic. Real solids have a continuous spectrum of phonon frequencies, as we learned in Chapter 3. Debye proposed using the phonon density of states g(ω) in a more realistic model.

4.2 Debye Approximation

Debye made a crucial simplification: approximate the phonon DOS with:

where ω_D is the Debye cutoff frequency, chosen so that:

(total of 3N modes for N atoms)

4.3 Debye Heat Capacity Formula

Total internal energy:

Defining the Debye temperature Θ_D = ℏω_D/k_B and x = ℏω/(k_BT):

The heat capacity is:

This integral defines the Debye function D(Θ_D/T).

4.4 Low and High Temperature Limits

High Temperature (T ≫ Θ_D):

✓ Recovers Dulong-Petit law

Low Temperature (T ≪ Θ_D):

When T ≪ Θ_D, the upper limit of integration becomes ∞, and the integral equals 4π⁴/15:

4.5 Physical Origin of T³ Law

Why does heat capacity scale as T³ at low temperature? Two factors:

1. Density of states: g(ω) ∝ ω² for acoustic phonons (3D Debye model)
2. Thermal occupation: Only phonons with ℏω ≲ k_BT are thermally excited. The number of such modes scales as (k_BT)³

Result: C_V ∝ (fraction of excited modes) × (energy per mode) ∝ T³ × T = T³

4.6 Debye Temperature Values

5. Comparison with Experimental Data

5.1 Heat Capacity Measurements

Experimental heat capacity measurements can be performed using calorimetry. The Debye model provides excellent agreement with experiment for most elemental solids, especially at low temperatures.

Key experimental observations:

• At T < Θ_D/10: C_V ∝ T³ holds very accurately
• At T ≈ Θ_D: Transition region between quantum and classical behavior
• At T > Θ_D: C_V → 3R (Dulong-Petit limit)

5.2 Deviations from Debye Model

The Debye model is not perfect. Deviations occur due to:

1. Electronic contribution: In metals, conduction electrons contribute C_el ∝ T at low temperature
2. Optical phonons: Materials with multiple atoms per unit cell have optical modes not captured by simple Debye model
3. Anharmonicity: At high T, anharmonic effects cause C_V to exceed 3R slightly

6. Python Implementation: Heat Capacity Calculations

6.1 Einstein Model Code

import numpy as np
import matplotlib.pyplot as plt
from scipy import integrate

def einstein_heat_capacity(T, theta_E, R=8.314):
  """
  Calculate heat capacity using Einstein model.

  Parameters:
  -----------
  T : float or array
    Temperature in Kelvin
  theta_E : float
    Einstein temperature in Kelvin
  R : float
    Gas constant (default: 8.314 J/(mol·K))

  Returns:
  --------
  C_V : float or array
    Molar heat capacity in J/(mol·K)
  """
  x = theta_E / T
  # Avoid division by zero or overflow
  x = np.clip(x, 1e-10, 100)

  exp_x = np.exp(x)
  C_V = 3 * R * (x**2) * exp_x / (exp_x - 1)**2

  return C_V

# Example: Calculate for Copper (theta_E ≈ 240 K)
T_range = np.linspace(10, 500, 200)
C_einstein = einstein_heat_capacity(T_range, theta_E=240)

plt.figure(figsize=(10, 6))
plt.plot(T_range, C_einstein, label='Einstein Model (Θ_E=240 K)', linewidth=2)
plt.axhline(y=3*8.314, color='r', linestyle='--', label='Dulong-Petit (3R)')
plt.xlabel('Temperature (K)', fontsize=12)
plt.ylabel('C_V (J/(mol·K))', fontsize=12)
plt.title('Einstein Model Heat Capacity', fontsize=14)
plt.legend(fontsize=11)
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

6.2 Debye Model Code

def debye_integrand(x):
  """Integrand for Debye heat capacity integral."""
  if x < 1e-10:
    return 0
  exp_x = np.exp(x)
  return (x**4 * exp_x) / (exp_x - 1)**2

def debye_heat_capacity(T, theta_D, R=8.314):
  """
  Calculate heat capacity using Debye model.

  Parameters:
  -----------
  T : float or array
    Temperature in Kelvin
  theta_D : float
    Debye temperature in Kelvin
  R : float
    Gas constant (default: 8.314 J/(mol·K))

  Returns:
  --------
  C_V : float or array
    Molar heat capacity in J/(mol·K)
  """
  # Handle array input
  T = np.atleast_1d(T)
  C_V = np.zeros_like(T, dtype=float)

  for i, temp in enumerate(T):
    if temp < 0.01:
      C_V[i] = 0
    elif temp > 10 * theta_D:
      # High temperature limit
      C_V[i] = 3 * R
    else:
      # Numerical integration
      x_max = theta_D / temp
      integral, _ = integrate.quad(debye_integrand, 0, x_max)
      C_V[i] = 9 * R * (temp / theta_D)**3 * integral

  return C_V if len(C_V) > 1 else C_V[0]

# Low-temperature T³ approximation
def debye_low_T_approx(T, theta_D, R=8.314):
  """
  Debye T³ law approximation for low temperatures.
  """
  return (12 * np.pi**4 / 5) * R * (T / theta_D)**3

# Example: Compare Debye model with T³ approximation
T_low = np.linspace(1, 100, 100)
theta_D_Cu = 343  # Copper

C_debye_full = debye_heat_capacity(T_low, theta_D_Cu)
C_debye_T3 = debye_low_T_approx(T_low, theta_D_Cu)

plt.figure(figsize=(10, 6))
plt.plot(T_low, C_debye_full, label='Full Debye Model', linewidth=2)
plt.plot(T_low, C_debye_T3, '--', label='T³ Approximation', linewidth=2)
plt.xlabel('Temperature (K)', fontsize=12)
plt.ylabel('C_V (J/(mol·K))', fontsize=12)
plt.title('Debye Model: Full vs T³ Approximation (Cu, Θ_D=343 K)', fontsize=14)
plt.legend(fontsize=11)
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

6.3 Comparing All Three Models

def compare_heat_capacity_models(theta_E=240, theta_D=343, material='Copper'):
  """
  Compare Dulong-Petit, Einstein, and Debye models.
  """
  T_range = np.logspace(0, 2.7, 200)  # 1 to ~500 K, log scale

  # Calculate heat capacities
  C_dulong_petit = 3 * 8.314 * np.ones_like(T_range)
  C_einstein = einstein_heat_capacity(T_range, theta_E)
  C_debye = debye_heat_capacity(T_range, theta_D)

  # Plotting
  fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))

  # Linear scale plot
  ax1.plot(T_range, C_dulong_petit, 'k--', label='Dulong-Petit', linewidth=2)
  ax1.plot(T_range, C_einstein, label=f'Einstein (Θ_E={theta_E} K)', linewidth=2)
  ax1.plot(T_range, C_debye, label=f'Debye (Θ_D={theta_D} K)', linewidth=2)
  ax1.set_xlabel('Temperature (K)', fontsize=12)
  ax1.set_ylabel('C_V (J/(mol·K))', fontsize=12)
  ax1.set_title(f'Heat Capacity Models - {material}', fontsize=14)
  ax1.legend(fontsize=11)
  ax1.grid(True, alpha=0.3)
  ax1.set_ylim(0, 30)

  # Log-log plot (better for low T)
  ax2.loglog(T_range, C_einstein, label=f'Einstein (Θ_E={theta_E} K)', linewidth=2)
  ax2.loglog(T_range, C_debye, label=f'Debye (Θ_D={theta_D} K)', linewidth=2)

  # Add T³ reference line
  T_ref = np.logspace(0, 1.5, 50)
  C_ref = 0.1 * (T_ref)**3
  ax2.loglog(T_ref, C_ref, 'k:', label='T³ reference', linewidth=1.5)

  ax2.set_xlabel('Temperature (K)', fontsize=12)
  ax2.set_ylabel('C_V (J/(mol·K))', fontsize=12)
  ax2.set_title(f'Log-Log Plot - {material}', fontsize=14)
  ax2.legend(fontsize=11)
  ax2.grid(True, alpha=0.3, which='both')

  plt.tight_layout()
  plt.show()

  # Print comparison at specific temperatures
  print(f"\n{material} Heat Capacity Comparison:")
  print(f"{'T (K)':<10} {'Dulong-Petit':<15} {'Einstein':<15} {'Debye':<15}")
  print("-" * 55)
  for T in [10, 50, 100, 200, 300, 500]:
    C_dp = 3 * 8.314
    C_e = einstein_heat_capacity(T, theta_E)
    C_d = debye_heat_capacity(T, theta_D)
    print(f"{T:<10} {C_dp:<15.2f} {C_e:<15.2f} {C_d:<15.2f}")

# Run comparison
compare_heat_capacity_models(theta_E=240, theta_D=343, material='Copper')

7. Thermal Expansion

7.1 The Origin of Thermal Expansion

Why do materials expand when heated? Surprisingly, the harmonic approximation predicts zero thermal expansion! In a purely harmonic potential, the average atomic position doesn't change with temperature—atoms just vibrate more vigorously about the same equilibrium point.

Thermal expansion arises from anharmonicity in the interatomic potential. Real potentials are asymmetric: it's easier to pull atoms apart than to push them together.

7.2 Anharmonic Potential

A realistic interatomic potential can be expanded as:

where:

• k is the harmonic force constant (k > 0)
• g is the cubic anharmonic term (typically g < 0)
• r₀ is the equilibrium separation at T = 0

The negative cubic term (g < 0) means the potential is softer (easier to stretch) at larger separations. As atoms vibrate with larger amplitude at higher T, they sample this asymmetric potential, leading to an increase in the average bond length.

7.3 Linear Thermal Expansion Coefficient

The linear thermal expansion coefficient is defined as:

where L is a linear dimension of the crystal.

For a 3D crystal, the volumetric thermal expansion coefficient is:

Thermal expansion is directly related to phonon anharmonicity and can be quantified using the Grüneisen parameter.

8. The Grüneisen Parameter

8.1 Definition

The Grüneisen parameter γ quantifies how phonon frequencies change when the crystal volume changes. For a phonon mode with frequency ω:

The mode Grüneisen parameter tells us: if we compress the crystal (reduce V), how much does the phonon frequency increase?

8.2 Physical Interpretation

Positive γ (typical): Compression increases phonon frequencies (atoms vibrate faster when pushed closer together). Most materials have γ ≈ 1–3.

Negative γ (rare): Some modes in certain materials have frequencies that decrease upon compression. This can lead to negative thermal expansion!

8.3 Average Grüneisen Parameter

For thermodynamic applications, we use an average Grüneisen parameter:

where C_V,i is the heat capacity contribution from mode i.

In the Debye approximation, assuming γ is constant for all modes:

8.4 Grüneisen Relation

The Grüneisen parameter connects thermal expansion to heat capacity through the fundamental relation:

where:

• α_V is the volumetric thermal expansion coefficient
• C_V is the heat capacity at constant volume
• B_T is the isothermal bulk modulus
• V is the volume

8.5 Typical Values

8.6 Temperature Dependence of Thermal Expansion

Since α_V ∝ C_V, thermal expansion follows the same temperature dependence as heat capacity:

• Low T: α_V ∝ T³ (follows Debye T³ law)
• High T: α_V ≈ constant (C_V → 3R)

This is observed experimentally: thermal expansion coefficients decrease dramatically as T → 0.

9. Anharmonicity and Its Role

9.1 Beyond the Harmonic Approximation

The harmonic approximation, while incredibly useful, fails to explain several important phenomena:

• Thermal expansion (requires anharmonicity)
• Thermal conductivity (phonon-phonon scattering needs anharmonicity)
• Temperature-dependent phonon frequencies
• Phonon lifetimes (infinite in harmonic approximation)

9.2 Anharmonic Phonon Interactions

In anharmonic theory, phonons can interact with each other. The cubic anharmonic term allows processes like:

• 3-phonon processes: One phonon splits into two, or two combine into one
• 4-phonon processes: Higher-order interactions (weaker)

Conservation laws apply:

where G is a reciprocal lattice vector (for Umklapp processes).

9.3 Impact on Thermal Properties

Thermal Conductivity:

Phonon-phonon scattering from anharmonicity limits thermal conductivity. At high T:

This is why diamond (very harmonic, weak anharmonicity) has extremely high thermal conductivity.

Thermal Expansion:

As discussed, anharmonicity is essential for thermal expansion through the Grüneisen mechanism.

Temperature-Dependent Phonon Frequencies:

Phonon frequencies shift with temperature due to anharmonic interactions:

This can be measured by temperature-dependent Raman spectroscopy or inelastic neutron scattering.

10. Summary

Connections to Other Chapters

This chapter completes the bridge from microscopic phonon theory (Chapters 1-3) to macroscopic thermal properties. In Chapter 5, we'll explore how these concepts apply to real materials and how experimental techniques measure phonon properties.

Exercises