---
title: The Sommerfeld Expansion and Electrons in Metals
module: Degenerate Fermi Gas
moduleNumber: 9
lessonNumber: 2
order: 902
summary: >
  Turning on a small temperature blurs the Fermi step over a shell of width
  $k_BT$ around $\epsilon_F$. The Sommerfeld expansion turns integrals over the
  Fermi function into a power series in $(k_BT/\epsilon_F)^2$, giving the shift of
  the chemical potential and a heat capacity linear in $T$. This resolves the old
  puzzle of the missing electronic heat capacity, predicts the combined
  $C=\gamma T+AT^3$ of a metal, and gives the temperature-independent Pauli
  paramagnetism of the electron gas.
topics: [Degenerate Fermi Gas]
sources:
  - book: Schroeder
    ref: "Ch. 7 — Quantum Statistics; §7.3 The Fermi Gas"
  - book: Ashcroft & Mermin
    ref: "Ch. 2 — The Sommerfeld Theory of Metals"
  - book: Pathria & Beale
    ref: "Ch. 8 — Ideal Fermi Systems; §8.1"
  - book: Reif
    ref: "Ch. 9 — Quantum Statistics of Ideal Gases; §9.17"
draft: false
---

At $T=0$ the Fermi gas is frozen into its ground state and contributes nothing to
the heat capacity. The classical equipartition estimate would give each electron
$\tfrac32 k_B$, so a metal should carry a large electronic heat capacity on top of
the lattice term. Experiment finds almost none at room temperature. The
resolution is that only a thin shell of electrons near the Fermi surface can
absorb thermal energy; the rest are locked below by fully occupied states above
them. The Sommerfeld expansion makes this quantitative.

## The thermally active shell

At finite temperature the occupation

$$
\bar n(\epsilon) = \frac{1}{e^{(\epsilon-\mu)/k_BT}+1}
$$

departs from the step only within a few $k_BT$ of $\mu$. Its energy derivative
$-\partial\bar n/\partial\epsilon$ is a peak of width $\sim k_BT$ centered on
$\mu$, normalized to unit area. Electrons more than a few $k_BT$ below the Fermi
energy cannot be excited: the states just above them are already full, so the
exclusion principle blocks any transition. Only the fraction
$\sim k_BT/\epsilon_F$ of electrons lying within the shell can move, and each that
does gains energy $\sim k_BT$.

$$
% caption: At finite $T$ the occupation softens over a shell of width $\sim k_BT$ around $\epsilon_F$; only electrons in the shaded band (blue) can be promoted into empty states just above.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6,0) node[right,black]{$E$};
\draw[->,black] (0,0)--(0,3.2) node[above,black]{occupation};
\fill[acc!16] (2.6,0) rectangle (3.8,2.4);
\draw[very thick] plot[domain=0:6,samples=120] (\x,{2.4/(1+exp((\x-3.2)/0.32))});
\draw[black,dashed] (0,2.4) node[left,black]{$1$} -- (2.6,2.4);
\draw[black,dashed] (3.2,0) node[below,black]{$E_F$} -- (3.2,2.4);
\node[acc] at (4.9,1.6) {width $k_BT$};
\draw[->,acc] (4.4,1.6)--(3.6,1.6);
\end{tikzpicture}
$$

This picture gives the heat capacity up to a numerical factor without any
integral. The thermal energy above the ground state is

$$
\Delta U \sim N\cdot\underbrace{\frac{k_BT}{\epsilon_F}}_{\text{active fraction}}
\cdot\underbrace{k_BT}_{\text{energy each}} = N k_B\,\frac{(k_BT)}{\epsilon_F}\,T,
$$

so the heat capacity is $C_V=\partial\Delta U/\partial T\sim N k_B\,T/T_F$, linear
in $T$ and smaller than the classical $\tfrac32 N k_B$ by the factor $T/T_F$. The
Sommerfeld expansion supplies the exact coefficient.

## The Sommerfeld expansion

Thermodynamic quantities are integrals of a smooth function $H(\epsilon)$ against
the Fermi function. For any $H$ that varies slowly on the scale $k_BT$ and
vanishes fast enough at $\epsilon=0$,[^sommerfeld]

$$
\int_0^\infty H(\epsilon)\,\bar n(\epsilon)\,\d\epsilon
= \int_0^\mu H(\epsilon)\,\d\epsilon
+ \frac{\pi^2}{6}(k_BT)^2\,H'(\mu)
+ \frac{7\pi^4}{360}(k_BT)^4\,H'''(\mu) + \cdots .
$$

The leading term is the zero-temperature result with $\epsilon_F$ replaced by
$\mu$; the corrections are even powers of $k_BT$ weighted by odd derivatives of
$H$ at the Fermi level. The expansion parameter is $(k_BT/\epsilon_F)^2$, which is
$\sim10^{-5}$ for a metal at room temperature.

[^sommerfeld]: The coefficients are moments of $-\partial\bar n/\partial\epsilon$;
$\pi^2/6$ and $7\pi^4/360$ are the values of the relevant Fermi integrals. Ashcroft
& Mermin, Ch. 2 appendix; Pathria & Beale, §8.1.

Fixing the density fixes $\mu(T)$ through $N=\int_0^\infty g(\epsilon)\bar
n(\epsilon)\,\d\epsilon$. Applying the expansion with $H=g$ and using
$g\propto\epsilon^{1/2}$ so that $g'/g = 1/2\epsilon$,

$$
\mu(T) = \epsilon_F\left[\,1 - \frac{\pi^2}{12}\left(\frac{k_BT}{\epsilon_F}\right)^2
+ \cdots\right].
$$

The chemical potential falls slightly below $\epsilon_F$ as the gas warms, because
the density of empty states just above $\epsilon_F$ exceeds the density of filled
states just below, and $\mu$ must drop to hold the particle number fixed.

## The electronic heat capacity

Applying the expansion to the energy, $H=\epsilon\,g(\epsilon)$, and eliminating
$\mu(T)$ in favor of $\epsilon_F$ gives

$$
U(T) = \frac{3}{5}N\epsilon_F\left[\,1 + \frac{5\pi^2}{12}\left(\frac{k_BT}{\epsilon_F}\right)^2
+ \cdots\right].
$$

Differentiating at fixed volume,

$$
C_{\mathrm{el}} = \left(\frac{\partial U}{\partial T}\right)_{V,N}
= \frac{\pi^2}{2}\,N k_B\,\frac{k_BT}{\epsilon_F}
= \frac{\pi^2}{2}\,N k_B\,\frac{T}{T_F}
= \frac{\pi^2}{3}\,k_B^2\,T\,g(\epsilon_F).
$$

The last form, written through $g(\epsilon_F)=3N/2\epsilon_F$, is the general
statement: the electronic heat capacity is set by the density of states at the
Fermi level. Define the **Sommerfeld coefficient**

$$
\gamma = \frac{\pi^2}{2}\,\frac{N k_B}{T_F} = \frac{\pi^2}{3}\,k_B^2\,g(\epsilon_F),
\qquad C_{\mathrm{el}} = \gamma T .
$$

The linear law replaces the constant classical prediction. At room temperature
$C_{\mathrm{el}}/\big(\tfrac32 N k_B\big)\sim T/T_F\sim10^{-2}$: the electronic
heat capacity is smaller than the classical value by two orders of magnitude,
which is why it went unnoticed before quantum statistics.

> **Worked example.** For copper, $T_F=8.2\times10^4\,\mathrm{K}$, so the molar
> Sommerfeld coefficient of the free-electron model is
>
> $$
> \gamma = \frac{\pi^2}{2}\frac{R}{T_F}
> = \frac{\pi^2}{2}\,\frac{8.314}{8.2\times10^4}
> = 5.0\times10^{-4}\,\mathrm{J\,mol^{-1}\,K^{-2}}.
> $$
>
> The measured value is $0.69\times10^{-3}\,\mathrm{J\,mol^{-1}\,K^{-2}}$. The
> free-electron model is right to within a factor of $1.4$; the discrepancy is
> absorbed into an effective mass $m^\ast\approx1.4\,m_e$ that accounts for the
> band structure and electron-lattice interactions the ideal gas ignores. At
> $T=300\,\mathrm{K}$ the electronic term $\gamma T\approx0.15\,\mathrm{J\,mol^{-1}\,K^{-1}}$
> is a percent of the lattice Dulong-Petit value $3R\approx25\,\mathrm{J\,mol^{-1}\,K^{-1}}$.

$$
% caption: The quantum electronic heat capacity $C_{\mathrm{el}}=\gamma T$ (solid) rises linearly from zero, far below the temperature-independent classical equipartition value $\tfrac32 N k_B$ (dashed).
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6,0) node[right,black]{$T$};
\draw[->,black] (0,0)--(0,4) node[above,black]{$C$};
\draw[black,thick,dashed] (0,3.0)--(5.6,3.0);
\node[black,left] at (5.6,3.35) {classical $\tfrac{3}{2}Nk_B$};
\draw[acc,very thick] (0,0)--(5.4,1.35);
\node[acc] at (4.0,0.62) {quantum};
\end{tikzpicture}
$$

## Electron and phonon contributions in a metal

At low temperature a metal carries two heat-capacity terms: the linear electronic
part $\gamma T$ and the Debye lattice part $A T^3$ from phonons. Their sum is

$$
C(T) = \gamma T + A T^3,\qquad A = \frac{12\pi^4}{5}\,\frac{N k_B}{\Theta_D^3},
$$

with $\Theta_D$ the Debye temperature. Dividing by $T$ linearizes the two
contributions,

$$
\frac{C}{T} = \gamma + A\,T^2 .
$$

Plotting $C/T$ against $T^2$ gives a straight line whose intercept is $\gamma$ and
whose slope is $A$. This extracts the density of states at the Fermi level from
the intercept and the Debye temperature from the slope, and is the standard way to
separate the electronic and lattice contributions in a low-temperature
measurement.[^cgamma]

[^cgamma]: The $C/T$-versus-$T^2$ construction and the measured $\gamma$ values are
in Ashcroft & Mermin, Ch. 2; Schroeder, §7.3, gives the electronic term.

$$
% caption: Plotting $C/T$ against $T^2$ turns $C=\gamma T+AT^3$ into a straight line; the intercept is the Sommerfeld coefficient $\gamma$ and the slope is the Debye coefficient $A$.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(6,0) node[right,black]{$T^2$};
\draw[->,black] (0,0)--(0,4) node[above,black]{$\frac{C}{T}$};
\draw[acc,very thick] (0,1.0)--(5.4,3.7);
\filldraw[acc] (0,1.0) circle (2.2pt);
\node[acc,below left] at (0.05,0.95) {intercept};
\draw[black] (3.0,{1.0+3.0*0.5})--(4.4,{1.0+3.0*0.5})--(4.4,{1.0+4.4*0.5});
\node[black,right] at (4.4,{1.0+3.7*0.5}) {slope $A$};
\end{tikzpicture}
$$

## Pauli paramagnetism

A magnetic field $B$ splits the electron gas by spin. An electron's energy shifts
by $\mp\mu_B B$ for spin aligned or anti-aligned with the field, where $\mu_B$ is
the Bohr magneton. The two spin populations then fill to a common chemical
potential, so the band lowered by the field holds more electrons than the raised
band. The imbalance is set by the density of states at the Fermi level: each spin
subband contributes $g(\epsilon_F)/2$, and its edge moves by $\mu_B B$, so the
number transferred is $\tfrac12 g(\epsilon_F)\,\mu_B B$ per band and the net
magnetization is

$$
M = \mu_B\big(N_\uparrow - N_\downarrow\big) = \mu_B^2\,g(\epsilon_F)\,B .
$$

The resulting **Pauli susceptibility**

$$
\chi_P = \mu_0\,\mu_B^2\,g(\epsilon_F)
$$

is independent of temperature, in contrast to the Curie law $\chi\propto1/T$ of
localized moments. The reason is the same shell argument: only the electrons
within $k_BT$ of the Fermi surface are free to flip, and their number is fixed by
$g(\epsilon_F)$ rather than by the total $N$. Localized-moment paramagnetism uses
every spin and therefore diverges as $T\to0$; the degenerate gas uses only the
Fermi-surface shell and stays finite.

$$
% caption: A field lowers the spin-up subband and raises the spin-down subband; filling both to the common Fermi level $E_F$ transfers electrons into the spin-up band, leaving a net moment fixed by $g(\epsilon_F)$.
\begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% spin up subband, shifted down: half-parabola opening left
\draw[acc,thick] plot[domain=0:2.6,samples=40] ({-1.6*sqrt(\x/2.6)},\x);
\fill[acc!16] plot[domain=0:2.1,samples=40] ({-1.6*sqrt(\x/2.6)},\x) -- (0,2.1) -- (0,0) -- cycle;
% spin down subband, shifted up
\draw[black,thick] plot[domain=0.5:3.1,samples=40] ({1.6*sqrt((\x-0.5)/2.6)},\x);
\fill[black] plot[domain=0.5:2.1,samples=40] ({1.6*sqrt((\x-0.5)/2.6)},\x) -- (0,2.1) -- (0,0.5) -- cycle;
\draw[black,dashed] (-2.0,2.1)--(2.0,2.1) node[right,black]{$E_F$};
\draw[->,black] (0,0)--(0,3.4) node[above,black]{$E$};
\node[acc,align=center] at (-1.9,2.9) {spin up};
\node[black,align=center] at (1.9,2.9) {spin down};
\node[black] at (0,-0.35) {$g(E)$ per spin};
\end{tikzpicture}
$$

A complementary orbital effect, **Landau diamagnetism**, arises from the
quantization of cyclotron orbits and gives a negative contribution
$\chi_L=-\tfrac13\chi_P$ for free electrons, so the net susceptibility of an ideal
gas is $\tfrac23\chi_P$. Band-structure effects modify both terms in real metals,
but the temperature-independent Pauli scale survives as the dominant feature of
the conduction-electron magnetism.[^landau]

[^landau]: The factor $-\tfrac13$ for free electrons is derived in Pathria & Beale,
§8.2A; Ashcroft & Mermin, Ch. 2, discuss the measured susceptibilities.

## Summary

- Finite temperature blurs the Fermi step over a shell of width $k_BT$ around
  $\epsilon_F$; only the fraction $\sim T/T_F$ of electrons in that shell respond
  thermally.
- The Sommerfeld expansion turns Fermi-function integrals into a series in
  $(k_BT/\epsilon_F)^2$, giving $\mu(T)=\epsilon_F[1-\tfrac{\pi^2}{12}(k_BT/\epsilon_F)^2]$
  and $U(T)=\tfrac35 N\epsilon_F[1+\tfrac{5\pi^2}{12}(k_BT/\epsilon_F)^2]$.
- The electronic heat capacity is linear, $C_{\mathrm{el}}=\gamma T$ with
  $\gamma=\tfrac{\pi^2}{3}k_B^2 g(\epsilon_F)$, resolving the missing-heat-capacity
  puzzle; the combined metal obeys $C=\gamma T+AT^3$, a line in $C/T$ versus $T^2$.
- Pauli paramagnetism gives a temperature-independent susceptibility
  $\chi_P=\mu_0\mu_B^2 g(\epsilon_F)$, with a Landau diamagnetic correction
  $-\tfrac13\chi_P$ for the free gas.