Einstein vs. Debye: Lattice Specific Heat

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome back to our exploration of lattice dynamics.

In the last lesson, we derived a general and powerful formula for the lattice specific heat, $C_{V}$ , showing that it's an integral over the phonon density of states, $g (ω)$ . We also discovered that the universal $ω^{2}$ dependence of $g (ω)$ at low frequencies for 3D crystals leads directly to the famous $T^{3}$ law for specific heat at low temperatures.

Today, we will investigate two landmark models that propose specific forms for $g (ω)$ to describe the specific heat across all temperatures. Our goal is to compare the Einstein and Debye models for lattice specific heat, deriving the T³ dependence in the low-temperature limit for the latter. This comparison will deepen our understanding of why a correct description of low-energy collective excitations is crucial for explaining the thermal properties of solids.

1. The Einstein Model: A First Quantum Attempt

The first quantum theory of specific heat was proposed by Einstein in 1907. It's based on a radical simplification of lattice vibrations.

Core Assumption: The crystal is treated as a collection of $N$ atoms, each vibrating independently as a quantum harmonic oscillator in three dimensions. Crucially, all these $3 N$ oscillators are assumed to vibrate at the same single frequency, $ω_{E}$ , known as the Einstein frequency.

This assumption implies a very simple density of states: all $3 N$ modes are concentrated at a single frequency.
$g (ω) = 3 N δ (ω - ω_{E})$
where $δ (ω - ω_{E})$ is the Dirac delta function.

To understand the consequences of this assumption, let's watch a detailed derivation of the Einstein model's heat capacity.

Introduction to Solid State Physics, Lecture 3: Einstein and Debye Models of a Solid

The following video by Prof. Sergey Frolov provides an excellent step-by-step derivation of the heat capacity in the Einstein model. It clearly explains the assumptions and analyzes the model's successes and failures.

Please watch the segment from 8:27 to 35:32. This covers the entire discussion of the Einstein model, from its core assumptions to the derivation of its specific heat and an analysis of its low and high-temperature limits.

As explained in the video, plugging our delta-function DOS into the general formula for specific heat from our last lesson,
$C_{V} = k_{B} \int_{0}^{\infty} d ω g (ω) (\frac{ℏ ω}{k _{B} T})^{2} \frac{e ^{ℏ ω / k_{B} T}}{( e ^{ℏ ω / k_{B} T} - 1 ) ^{2}}$
gives the Einstein model's specific heat:
$C_{V} (T) = 3 N k_{B} (\frac{ℏ ω _{E}}{k _{B} T})^{2} \frac{e ^{ℏ ω_{E} / k_{B} T}}{( e ^{ℏ ω_{E} / k_{B} T} - 1 ) ^{2}}$

Let's analyze its behavior in the two temperature limits:

High-Temperature Limit ( $k_{B} T ≫ ℏ ω_{E}$ ):
In this limit, the argument of the exponential $x = ℏ ω_{E} / k_{B} T \to 0$ . Using the approximation $e^{x} \approx 1 + x$ , the term $(e^{x} - 1)^{2} \approx x^{2}$ . The numerator $x^{2} e^{x} \approx x^{2}$ . The expression simplifies to $C_{V} \to 3 N k_{B}$ . This correctly recovers the classical Dulong-Petit law, a significant success.
Low-Temperature Limit ( $k_{B} T ≪ ℏ ω_{E}$ ):
Here, $x ≫ 1$ , so $e^{x} - 1 \approx e^{x}$ . The specific heat becomes:
$C_{V} \approx 3 N k_{B} x^{2} \frac{e ^{x}}{( e ^{x} ) ^{2}} = 3 N k_{B} x^{2} e^{- x} = 3 N k_{B} (\frac{ℏ ω _{E}}{k _{B} T})^{2} e^{- ℏ ω_{E} / k_{B} T}$
The specific heat goes to zero as $T \to 0$ , which is qualitatively correct. However, it does so exponentially. This contradicts the experimentally observed $T^{3}$ dependence.

The physical reason for this failure is clear: the model assumes that the lowest-energy excitation available has a finite energy $ℏ ω_{E}$ . At low temperatures ( $k_{B} T ≪ ℏ ω_{E}$ ), there is insufficient thermal energy to excite even this first mode, leading to an exponential "freezing out" of the heat capacity. The model completely neglects the existence of low-frequency collective vibrations (acoustic phonons) that can be excited with arbitrarily small energy.

2. The Debye Model: Correcting the Low-Energy Picture

In 1912, Peter Debye proposed a more realistic model by focusing on the collective vibrational modes of the solid, not independent atomic vibrations.

Core Assumptions:

The solid is treated as a continuous elastic medium. This means the model quantizes the sound waves (phonons) propagating through the solid.
The dispersion relation for these sound waves is assumed to be linear for all three polarizations: $ω = v_{s} ∣ k ∣$ , where $v_{s}$ is an average speed of sound.
The total number of vibrational modes is finite and equal to $3 N$ , just like in the Einstein model. This is inconsistent with a true continuum (which would have infinite modes), so Debye introduced a crucial "fix": he imposed a cutoff frequency, $ω_{D}$ , such that there are no modes with $ω > ω_{D}$ .

The first two assumptions lead directly to the low-frequency density of states we derived in the last lesson: $g (ω) \propto ω^{2}$ . The third assumption determines the cutoff.

The following text provides a clear derivation of the Debye model, including the low-temperature $T^{3}$ law.

Debye model

The online textbook from Quantum Tinkerer offers a very clear, modern presentation of the Debye model. It methodically derives the key results we need.

Please read the sections titled 'Low temperatures' and 'Debye's interpolation for medium T'. The first section re-derives the T³ law, which is central to today's lesson. The second section explains the concept of the Debye frequency (ω_D) and Debye temperature (T_D) and presents the full integral for the heat capacity.

Let's consolidate the key steps from the reading.

Deriving the Debye Cutoff Frequency $ω_{D}$

The cutoff $ω_{D}$ is determined by the condition that the total number of modes equals $3 N$ :
$\int_{0}^{ω_{D}} g (ω) d ω = 3 N$
Using the DOS for a 3D continuum, $g (ω) = \frac{3 V}{2 π ^{2} v _{s}^{3}} ω^{2}$ :
$\int_{0}^{ω_{D}} \frac{3 V}{2 π ^{2} v _{s}^{3}} ω^{2} d ω = \frac{3 V}{2 π ^{2} v _{s}^{3}} \frac{ω _{D}^{3}}{3} = \frac{V ω _{D}^{3}}{2 π ^{2} v _{s}^{3}} = 3 N$
Solving for $ω_{D}$ , we get the Debye frequency:
$ω_{D} = v_{s} (6 π^{2} \frac{N}{V})^{1/3}$
Notice that $ω_{D}$ depends on the material's properties: the speed of sound $v_{s}$ and the atomic density $N / V$ .

With this, the Debye density of states is:
$g (ω) = {\frac{9 N}{ω _{D}^{3}} ω^{2} 0 for ω \leq ω_{D} for ω > ω_{D}$
The prefactor is written in this form to ensure that $\int_{0}^{ω_{D}} g (ω) d ω = 3 N$ .

Specific Heat: The Low-Temperature $T^{3}$ Law

The heat capacity is found by integrating over this DOS. It's convenient to define the Debye Temperature $T_{D} = ℏ ω_{D} / k_{B}$ .
$C_{V} (T) = 9 N k_{B} (\frac{T}{T _{D}})^{3} \int_{0}^{T_{D} / T} \frac{x ^{4} e ^{x}}{( e ^{x} - 1 ) ^{2}} d x where x = \frac{ℏ ω}{k _{B} T}$

At low temperatures ( $T ≪ T_{D}$ ), the upper limit of the integral $T_{D} / T \to \infty$ . This is physically justified because the Bose-Einstein factor $(e^{x} - 1)^{- 1}$ strongly suppresses the contribution from high-energy phonons ( $ℏ ω ≫ k_{B} T$ ), so extending the integral limit from $ω_{D}$ to $\infty$ makes no practical difference.
$C_{V} \approx 9 N k_{B} (\frac{T}{T _{D}})^{3} \int_{0}^{\infty} \frac{x ^{4} e ^{x}}{( e ^{x} - 1 ) ^{2}} d x$
The integral is a standard definite integral equal to $\frac{4 π ^{4}}{15}$ . Substituting this gives:
$C_{V} = 9 N k_{B} (\frac{T}{T _{D}})^{3} (\frac{4 π ^{4}}{15}) = \frac{12 π ^{4}}{5} N k_{B} (\frac{T}{T _{D}})^{3}$
This is the celebrated Debye T³ law. The model successfully reproduces the correct low-temperature behavior because it correctly incorporates the density of low-energy acoustic modes.

High-Temperature Limit

At high temperatures ( $T ≫ T_{D}$ ), the integration variable $x = ℏ ω / k_{B} T ≪ 1$ . The integrand can be approximated: $\frac{x ^{4} e ^{x}}{( e ^{x} - 1 ) ^{2}} \approx \frac{x ^{4} ( 1 )}{( x ) ^{2}} = x^{2}$ .
$C_{V} \approx 9 N k_{B} (\frac{T}{T _{D}})^{3} \int_{0}^{T_{D} / T} x^{2} d x = 9 N k_{B} (\frac{T}{T _{D}})^{3} [\frac{x ^{3}}{3}]_{0}^{T_{D} / T}$
$C_{V} \approx 9 N k_{B} (\frac{T}{T _{D}})^{3} \frac{1}{3} (\frac{T _{D}}{T})^{3} = 3 N k_{B}$
The Debye model also correctly recovers the Dulong-Petit law at high temperatures.

Test your understanding!

The Debye temperature $T_{D}$ is a measure of the maximum phonon frequency in a solid. Diamond has a very high Debye temperature ( $T_{D} \approx 2230 K$ ), while lead has a very low one ( $T_{D} \approx 105 K$ ). Based on the physics of the models, why do you think this is the case? Relate it to material properties.

Show answer

The Debye frequency is given by $ω_{D} = v_{s} (6 π^{2} N / V)^{1/3}$ . It depends on two key properties:

Speed of sound ( $v_{s}$ ): This is related to the stiffness of the atomic bonds and the mass of the atoms. Diamond is extremely rigid (strong covalent bonds) and made of light carbon atoms, leading to a very high speed of sound. Lead is soft (weak metallic bonds) and has heavy atoms, leading to a low speed of sound.
Atomic density ( $N / V$ ): Diamond also has a very high atomic density.

The combination of a very high $v_{s}$ and high $N / V$ gives diamond a very large $ω_{D}$ and thus a very high $T_{D}$ . Physically, this means you need much more thermal energy to excite the highest-frequency vibrational modes in diamond compared to lead. This is also why diamond's specific heat deviates from the Dulong-Petit law even at room temperature, while lead's follows it quite well.

3. Comparison and Conclusion

The Debye model represents a significant improvement over the Einstein model. Its success lies in correctly modeling the low-energy excitations as collective sound waves.

Comparison of Debye and Einstein Models for Specific Heat — This plot compares the heat capacity predicted by the Debye model (blue line) and the Einstein model (orange line) against experimental data for silver. The Debye model provides a much better fit, especially in the low-temperature regime where the T³ dependence is crucial.

Limitations of the Debye Model:
Despite its success, the Debye model is still an approximation:

The assumption of a linear dispersion $ω = v_{s} q$ is only valid for long-wavelength acoustic phonons. Real dispersion curves flatten near the Brillouin zone boundary.
The sharp cutoff at $ω_{D}$ is an artificial construct. A real DOS does not simply stop but has complex features (van Hove singularities) related to the actual crystal structure.
It uses an average speed of sound, ignoring the difference between longitudinal and transverse modes.

However, because the Debye model gets the two most important features right—the correct $T^{3}$ dependence at low temperatures and the correct Dulong-Petit value at high temperatures—it serves as an excellent and widely used approximation.

Summary of Key Takeaways

Einstein Model: Assumes independent atoms oscillating at a single frequency $ω_{E}$ . It correctly predicts $C_{V} \to 3 N k_{B}$ at high T but fails at low T, where it predicts an incorrect exponential decay.
Debye Model: Treats lattice vibrations as collective sound waves (phonons) in a continuous medium. It assumes a linear dispersion up to a cutoff frequency $ω_{D}$ that ensures the correct number of modes.
The T³ Law: The Debye model's success stems from its $g (ω) \propto ω^{2}$ density of states for low-energy acoustic modes, which correctly yields the $C_{V} \propto T^{3}$ law at low temperatures.
The Debye model provides a robust interpolation that accurately describes the lattice specific heat in both the low and high-temperature limits, making it a cornerstone of solid-state physics.

Preview of the Next Lesson

Both the Einstein and Debye models propose idealized forms for the phonon density of states. But how do we determine the actual phonon dispersion curves $ω (q)$ and the true $g (ω)$ for a real crystal? In the next lesson, we will explore the primary experimental technique for this: inelastic neutron scattering. This will directly connect our theoretical framework of phonons to measurable experimental data, a link that is crucial to your goal of connecting theory with experimental work.