Sommerfeld Expansion and Electronic Heat Capacity

In our last lesson, we explored the physics of a Fermi gas at the extreme limit of absolute zero, revealing the powerful quantum mechanical effect of degeneracy pressure and its role in stabilizing white dwarf stars. We saw that at T=0, the Fermi-Dirac distribution becomes a sharp step function, simplifying our calculations immensely. However, most real systems, like the sea of electrons in a metal, exist at low but non-zero temperatures. At these temperatures, the distribution function is no longer a perfect step, and the exact integrals for energy and particle number become intractable.

Today, we will bridge the gap between the T=0 idealization and finite-temperature reality. Our objective is to perform the Sommerfeld expansion to derive the low-temperature electronic heat capacity of metals, showing that it is proportional to temperature ( $C_{V} \propto T$ ). This result was one of the early triumphs of quantum statistics, explaining experimental observations that classical physics could not. The mathematical technique we will develop is a cornerstone for analyzing degenerate Fermi systems.

1. The Challenge of Finite Temperature

Recall the general expressions for the total number of particles $N$ and internal energy $U$ :

N = \int_{0}^{\infty} g (ε) f (ε) d ε and U = \int_{0}^{\infty} ε g (ε) f (ε) d ε

where $g (ε)$ is the density of states and $f (ε) = (e^{(ε - μ) / (k_{B} T)} + 1)^{- 1}$ is the Fermi-Dirac distribution.

The difficulty lies entirely in the form of $f (ε)$ . At finite $T$ , this function smoothly transitions from 1 to 0 in a "smeared" region of width $\sim k_{B} T$ around the chemical potential $μ$ . The essence of the Sommerfeld expansion is to recognize that for low temperatures ( $k_{B} T ≪ μ$ ), the temperature dependence of these integrals comes only from this small, smeared region near $μ$ . The expansion systematically calculates the corrections to the T=0 result arising from this smearing.

The integrals are of a general form:

I = \int_{0}^{\infty} H (ε) f (ε) d ε

where $H (ε)$ is some well-behaved function (e.g., $g (ε)$ for the particle number, or $ε g (ε)$ for the energy). The Sommerfeld expansion provides an asymptotic series for this integral in powers of $k_{B} T / μ$ . The formula is:

\int_{- \infty}^{\infty} H (ε) f (ε) d ε = \int_{- \infty}^{μ} H (ε) d ε + \frac{π ^{2}}{6} (k_{B} T)^{2} H^{'} (μ) + \frac{7 π ^{4}}{360} (k_{B} T)^{4} H^{'''} (μ) + \dots

Note that the lower limit is extended to $- \infty$ , which is permissible since $H (ε)$ is typically zero for $ε < 0$ .

Given your interest in mathematical rigor, you might find it valuable to see how this expansion is formally derived. It involves a clever change of variables, Taylor expansion, and evaluation of standard integrals related to the Riemann zeta function.

[PDF] 1. Power Series Expansion of the Fermi-Dirac Integral

This document provides a detailed, step-by-step derivation of the asymptotic expansion that forms the basis of the Sommerfeld expansion.

Please read the section "Asymptotic Expansion of the Fermi-Dirac Integral" (starting on page 2). Follow the steps from eq. (3) onwards. The key steps are: Subtracting the T=0 step-function behavior to isolate the correction term, h_n(y). Changing variables to w = x - y and expanding (w+y)^{n-1} in a power series. Interchanging the sum and integral and evaluating the resulting integrals over w. The final boxed result for f_n(y) is equivalent to the Sommerfeld expansion formula.

2. Low-Temperature Properties of a Fermi Gas

With the Sommerfeld expansion formula in hand, we can now compute the leading-order temperature corrections to the chemical potential and internal energy. The following problem solution sheet from ETH Zurich provides a concise and rigorous path through this calculation.

[PDF] Statistical Physics. Solutions Sheet 5. - ITP Lecture Archive

This document applies the Sommerfeld expansion directly to the integrals for particle number and energy.

In "Exercise 2", start by reviewing part (a) which sets up the integrals for particle and energy densities in terms of the density of states g(\varepsilon). Then, focus on the Solution to part (b). Follow how the Sommerfeld expansion formula (8) is applied to n and u, leading to equations (S.26) and (S.27). Note the use of the hint: the integral up to \mu is itself expanded around the T=0 Fermi energy \varepsilon_F, since \mu will only deviate from \varepsilon_F by a small amount proportional to T^2. This leads to the key approximate expressions for n and u in equations (S.28) and (S.29).

Let's break down the logic of that derivation.

Step 1: Find the temperature dependence of the chemical potential, $μ (T)$

We start by applying the expansion to the particle number density $n = N / V$ , which must remain constant as we turn on the temperature.

n = \int_{0}^{\infty} g (ε) f (ε) d ε \approx \int_{0}^{μ} g (ε) d ε + \frac{π ^{2}}{6} (k_{B} T)^{2} g^{'} (μ)

At $T = 0$ , we have $n = \int_{0}^{ε_{F}} g (ε) d ε$ . Since $n$ must be the same at finite $T$ , and $μ \approx ε_{F}$ , we can expand the integral around $ε_{F}$ :

\int_{0}^{μ} g (ε) d ε \approx \int_{0}^{ε_{F}} g (ε) d ε + (μ - ε_{F}) g (ε_{F}) = n + (μ - ε_{F}) g (ε_{F})

Substituting this back and setting $g^{'} (μ) \approx g^{'} (ε_{F})$ , we get:

n \approx n + (μ - ε_{F}) g (ε_{F}) + \frac{π ^{2}}{6} (k_{B} T)^{2} g^{'} (ε_{F})

For this equation to hold, the correction terms must cancel. This gives us the chemical potential's temperature dependence:

μ (T) \approx ε_{F} - \frac{π ^{2}}{6} (k_{B} T)^{2} \frac{g ^{'} ( ε _{F} )}{g ( ε _{F} )}

The chemical potential decreases slightly as temperature increases (assuming $g^{'} (ε_{F}) > 0$ , which is true for a free gas). This makes physical sense: as the distribution smears, states above $ε_{F}$ become populated. To keep the total number of particles constant, the chemical potential must shift downwards.

Step 2: Find the internal energy, $U (T)$

Now we apply the same procedure to the internal energy density $u = U / V$ .

u = \int_{0}^{\infty} ε g (ε) f (ε) d ε \approx \int_{0}^{μ} ε g (ε) d ε + \frac{π ^{2}}{6} (k_{B} T)^{2} \frac{d}{d μ} (μg (μ))

Evaluating the derivative and expanding the integral around $ε_{F}$ , we get:

u \approx [\int_{0}^{ε_{F}} ε g (ε) d ε + (μ - ε_{F}) ε_{F} g (ε_{F})] + \frac{π ^{2}}{6} (k_{B} T)^{2} [g (ε_{F}) + ε_{F} g^{'} (ε_{F})]

The term in the square brackets is $u_{0} + (μ - ε_{F}) ε_{F} g (ε_{F})$ , where $u_{0}$ is the energy density at $T = 0$ . Substituting our expression for $(μ - ε_{F})$ from Step 1, we find that the terms involving $g^{'} (ε_{F})$ cancel out, leaving a remarkably simple result:

u (T) \approx u_{0} + \frac{π ^{2}}{6} (k_{B} T)^{2} g (ε_{F})

The internal energy increases quadratically with temperature.

3. Electronic Heat Capacity of Metals

We are now in a position to calculate the heat capacity. The electronic specific heat (heat capacity per unit volume) at constant volume is $c_{V} = (\partial u / \partial T)_{n}$ . Differentiating our expression for $u (T)$ gives the general result:

c_{V} = \frac{\partial}{\partial T} (u_{0} + \frac{π ^{2}}{6} g (ε_{F}) (k_{B} T)^{2}) = \frac{π ^{2}}{3} g (ε_{F}) k_{B}^{2} T

This is a central result of the free electron model. It predicts that the electronic contribution to the heat capacity is linear in temperature. This linearity is a direct consequence of the shape of the Fermi-Dirac distribution; only electrons within a narrow energy window of width $\sim k_{B} T$ around the Fermi surface can be excited. The number of such electrons is proportional to $T$ , and each absorbs an energy of about $k_{B} T$ , so the total energy absorbed is proportional to $T^{2}$ , and the heat capacity is proportional to $T$ .

To get a concrete formula for a simple metal, we can substitute the density of states for a 3D free electron gas, which you know is $g (ε) = C ε^{1/2}$ .

[PDF] Statistical Physics. Solutions Sheet 5. - ITP Lecture Archive

This final part of the document applies the general results to the specific case of the free electron gas.

Read through the Solutions for parts (c) and (d). Part (c) consolidates the derivation of \mu and gives the general result for c_v that we just discussed: c_v \propto T g(\varepsilon_F). Notice the comment that heat capacity measurements are a direct probe of the density of states at the Fermi energy. Part (d) substitutes the density of states for a free Fermi gas, g(\varepsilon), into the general formula. Follow the algebra that leads to the final expression for the specific heat, equation (S.38). It also provides the crucial comparison to the classical result.

By expressing $g (ε_{F})$ in terms of the particle density $n$ and the Fermi energy $ε_{F}$ (using $g (ε_{F}) = \frac{3}{2} n / ε_{F}$ ), we arrive at the final result for the heat capacity per volume:

c_{V} = \frac{π ^{2}}{2} n k_{B} (\frac{k _{B} T}{ε _{F}})

Defining the Fermi Temperature $T_{F} = ε_{F} / k_{B}$ , we can write the molar heat capacity $C_{V, m}$ (where $N_{A}$ is Avogadro's number and $R = N_{A} k_{B}$ ) as:

C_{V, m} = \frac{π ^{2}}{2} R (\frac{T}{T _{F}}) = γ T

This is the famous linear temperature dependence. For typical metals, $T_{F}$ is on the order of $1 0^{4} - 1 0^{5}$ K, so at room temperature ( $\sim 300$ K), the factor $T / T_{F}$ is very small ( $\sim 1 0^{- 2}$ ). This explains why the electronic contribution to the heat capacity is much smaller than the classical prediction of $\frac{3}{2} R$ and is also much smaller than the contribution from lattice vibrations (phonons), except at very low temperatures.

This graph compares the theoretical predictions for the electronic specific heat of Aluminium with experimental data. The dashed line ($C_e = \gamma T_e$) is the first-order Sommerfeld expansion result we just derived. It shows excellent agreement with the published data at low temperatures. The solid line is a more complete analytical solution, and the dotted line includes higher-order corrections. The plot powerfully validates our low-temperature approximation.

Conclusion

In this lesson, we have moved beyond the T=0 approximation and developed a powerful tool, the Sommerfeld expansion, to describe the behavior of Fermi gases at low temperatures. This allowed us to quantitatively explain a key property of metals that baffled classical physics.

Key Takeaways:

The Sommerfeld expansion is an asymptotic expansion that allows us to calculate thermodynamic properties of a highly degenerate Fermi gas at low temperatures ( $k_{B} T ≪ ε_{F}$ ).
Applying the expansion, we found that the chemical potential decreases quadratically with temperature, $μ (T) \approx ε_{F} - O (T^{2})$ .
The internal energy increases quadratically from its T=0 value, $U (T) \approx U_{0} + O (T^{2})$ .
The electronic heat capacity of a metal is linearly proportional to temperature, $C_{V} = γ T$ , where $γ \propto g (ε_{F})$ , the density of states at the Fermi energy.
This linear dependence arises because the Pauli exclusion principle only permits electrons within an energy range $\sim k_{B} T$ of the Fermi surface to participate in thermal processes.

We have now seen how the Fermi-Dirac statistics govern both the immense pressures inside stars and the subtle thermal properties of everyday metals. In our next lesson, we will apply the same framework to understand another magnetic property of the electron gas: Pauli paramagnetism.

Recommended resources

1. The Challenge of Finite Temperature

2. Low-Temperature Properties of a Fermi Gas

3. Electronic Heat Capacity of Metals

Conclusion