Pauli Paramagnetism: Free Electron Gas Susceptibility

Welcome to our next lesson. In our previous session, we delved into the low-temperature behavior of the electron gas using the Sommerfeld expansion. We discovered that the linear temperature dependence of the electronic heat capacity in metals, $C_{V} \propto T$ , is a direct consequence of the Pauli exclusion principle, which dictates that only electrons near the Fermi surface can participate in thermal excitations.

This lesson builds directly on that foundation. We will investigate another key property of the free electron gas that is profoundly shaped by the Fermi surface: its magnetic response. Our objective is to explain Pauli paramagnetism by calculating the low-field magnetic susceptibility of a free electron gas. You will see how the same principles that govern heat capacity also explain why the magnetism of simple metals is weak and largely independent of temperature, in stark contrast to the behavior of classical magnetic moments.

1. Electron Spin and Magnetic Fields

An electron possesses an intrinsic quantum property called spin, which gives it a magnetic dipole moment, $μ$ . The magnitude of this moment is given by the Bohr magneton, $μ_{B} = e ℏ/2 m_{e}$ . Due to the electron's negative charge, its magnetic moment points in the direction opposite to its spin angular momentum.

When we place an electron gas in an external magnetic field, $B$ , the energy of each electron is shifted by the Zeeman effect. The interaction energy is $U = - μ \cdot B$ . Let's align our coordinate system so that $B = B z^$ .

An electron with spin "up" (spin angular momentum parallel to $B$ ) has its magnetic moment pointing down (antiparallel to $B$ ). Its energy is increased by $+ μ_{B} B$ .
An electron with spin "down" (spin angular momentum antiparallel to $B$ ) has its magnetic moment pointing up (parallel to $B$ ). Its energy is decreased by $- μ_{B} B$ .

The following video provides a helpful overview of the origin of magnetism from electron spin and this energy-level splitting.

Magnetism - Statistical Physics - University Physics

This video from the channel Pazzy Boardman provides a clear, qualitative introduction to paramagnetism, linking electron structure to magnetic moments and deriving the Zeeman energy splitting.

Watch the initial segment explaining how unpaired electron spins give rise to magnetic dipole moments: origin of magnetism. Next, focus on how these moments interact with an external magnetic field and the resulting competition between energy minimization and thermal entropy: aligning with a field. Finally, pay close attention to the derivation of the Zeeman energy levels. The video establishes the two-state system we will be analyzing: Zeeman energy. Note that the video's sign convention for spin-up/down energy may differ, but the key concept of an energy splitting of 2\mu_B B is universal.

As a result of this energy shift, the continuous band of electron states splits into two sub-bands: one for spin-up electrons, shifted up in energy by $μ_{B} B$ , and one for spin-down electrons, shifted down by $μ_{B} B$ . The total energy of an electron with kinetic energy $ε_{k}$ is now $ε = ε_{k} \pm μ_{B} B$ .

2. Magnetization at Zero Temperature

Let's first consider the situation at $T = 0$ . In the absence of a magnetic field ( $B = 0$ ), the spin-up and spin-down states are degenerate. The electrons fill both sub-bands up to the same Fermi energy, $ε_{F}$ , and the number of spin-up electrons ( $N_{↑}$ ) equals the number of spin-down electrons ( $N_{↓}$ ). The net magnetization is zero.

Now, we apply a small magnetic field $B$ . The spin-down sub-band moves to lower energies, and the spin-up sub-band moves to higher energies. To maintain a single, uniform chemical potential (which at $T = 0$ is the Fermi energy), electrons in the highest-energy spin-up states will flip their spin and occupy the now-available lower-energy spin-down states.

This process continues until the topmost filled states in both sub-bands are at the same energy level, $μ$ . This results in an excess of spin-down electrons: $N_{↓} > N_{↑}$ .

This diagram visualizes the effect of a magnetic field on the electron gas at T=0. The parabolic curves represent the density of states for "Moments up" (spin-down electrons) and "Moments down" (spin-up electrons). Initially (solid line), both are filled to the same Fermi energy $\varepsilon_F$. When the field is applied, the spin-down band shifts left (lower energy) and the spin-up band shifts right (higher energy). To maintain a new, common chemical potential (dashed line), more spin-down states become occupied, creating a net magnetic moment.

This population imbalance creates a net magnetic moment parallel to the field. The total magnetization $M$ is the net magnetic moment per unit volume, which is given by:

M = μ_{B} (n_{↓} - n_{↑})

where $n_{↑} = N_{↑} / V$ and $n_{↓} = N_{↓} / V$ are the number densities.

For a small applied field ( $μ_{B} B ≪ ε_{F}$ ), we can calculate this imbalance. The number of spin-up electrons that flip their spin is approximately the number of states in an energy slice of width $μ_{B} B$ at the top of the Fermi sea. The density of states for a single spin species is $\frac{1}{2} g (ε)$ .
The number of electrons that flip from up to down is roughly $(\frac{1}{2} g (ε_{F})) \times (μ_{B} B)$ . This creates a deficit of this amount in the spin-up population and a surplus of the same amount in the spin-down population.

Therefore, the difference in populations is:

N_{↓} - N_{↑} \approx 2 \times (\frac{1}{2} g (ε_{F}) μ_{B} B) = g (ε_{F}) μ_{B} B

The magnetization is then:

M = \frac{μ _{B}}{V} (N_{↓} - N_{↑}) \approx \frac{μ _{B}^{2} g ( ε _{F} )}{V} B

The magnetic susceptibility, $χ$ , measures the material's response to the field. For a paramagnetic material, it's defined as $χ = \partial M / \partial B$ (in some conventions; others use $χ = μ_{0} M / B$ ). Using the derivative definition, we find:

χ_{P} = μ_{B}^{2} \frac{g ( ε _{F} )}{V}

This is the celebrated result for Pauli paramagnetism. It states that the magnetic susceptibility of a free electron gas at low temperature is a positive constant, determined entirely by the density of states at the Fermi energy. This is a purely quantum mechanical effect, driven by the Pauli exclusion principle. Only the small fraction of electrons at the Fermi surface are free to realign their spins; the vast majority are "frozen" in the Fermi sea.

3. Rigorous Calculation

Let's formalize the argument above. The following text by David Tong provides a concise derivation. We will follow his approach, which uses the integral expressions for particle number that you are familiar with from our previous work with Fermi gases.

3. Quantum Gases

This section of David Tong's lecture notes provides a clear, rigorous derivation of the magnetic susceptibility for a Fermi gas. It contrasts the low-temperature Pauli paramagnetism with the high-temperature classical Curie's Law.

Begin reading section 3.6.6, "Pauli Paramagnetism". First, understand how the number of spin-up (N_\uparrow) and spin-down (N_\downarrow) electrons are expressed as separate integrals: these equations. Briefly look at the high-temperature derivation, which recovers the classical Curie's Law, where \chi \propto 1/T. This serves as a crucial point of contrast. Now, focus on the main low-temperature calculation. Follow the argument starting from this paragraph. The key step is to use the T=0 approximation for the integrals and expand the result for a small magnetic field B. Finally, see how this result is connected back to the density of states at the Fermi energy, leading to the final expressions for magnetization and susceptibility.

The derivation shows that by taking the T=0 energy expressions for the spin-up and spin-down populations and expanding for small $B$ , we arrive precisely at $M = μ_{B}^{2} g (ε_{F}) B$ , confirming our intuitive argument.

4. Finite Temperature Correction

Your background and interest in formal derivations make it worthwhile to see how this result is corrected at low but finite temperatures. As we saw with heat capacity, the Sommerfeld expansion is the tool for this. A more advanced calculation using the grand canonical potential and the Sommerfeld expansion reveals the first temperature-dependent correction.

Fermi-Dirac statistics: Pauli paramagnetism

This document by M.S. Suzuki provides two derivations. The first is a clear graphical and algebraic argument at T=0. The second is a more formal grand canonical treatment that reveals the first-order temperature correction.

First, quickly review the section "1. Pauli paramagnetism". The diagram and the derivation culminating in the T=0 result provide an excellent summary of what we've covered so far. Next, delve into section "2. Pauli paramagnetism (analytical approach)". You do not need to reproduce every step, but trace the logic. The goal is to find the total particle numbers N_+ and N_- as functions of T and B. By requiring N = N_+ + N_- to be constant, one finds the chemical potential \mu(T, B). Substituting this back into the expression for the magnetization, M = \mu_B (N_+ - N_-), yields the final result. Focus on the final expression for the Pauli susceptibility as a function of temperature.

The final result from this more rigorous calculation is:

χ_{P} (T) = μ_{B}^{2} \frac{g ( ε _{F} )}{V} [1 - \frac{π ^{2}}{12} (\frac{k _{B} T}{ε _{F}})^{2} + \dots]

This confirms that at $T = 0$ , the susceptibility is $μ_{B}^{2} g (ε_{F}) / V$ . It also shows that for low temperatures, there is a small, negative correction that scales with $T^{2}$ . This weak temperature dependence is a hallmark of degenerate Fermi systems and stands in stark contrast to the $1/ T$ dependence of classical (Curie) paramagnetism, where all magnetic moments are free to align with the field.

Conclusion

In this lesson, we have applied our understanding of the degenerate Fermi gas to explain the magnetic properties of simple metals. We have seen how the Pauli exclusion principle, which dominates the physics near the Fermi surface, leads to a weak and nearly temperature-independent form of magnetism known as Pauli paramagnetism.

Key Takeaways:

An external magnetic field splits the electron gas into two sub-bands (spin-up and spin-down) separated by an energy $2 μ_{B} B$ .
To minimize energy, electrons near the Fermi surface flip their spin, creating a net excess of spins whose magnetic moments are aligned with the field.
The resulting magnetization is proportional to the applied field $B$ and the density of states at the Fermi energy, $g (ε_{F})$ .
The Pauli paramagnetic susceptibility, $χ_{P} = μ_{B}^{2} g (ε_{F}) / V$ , is a positive constant at low temperatures. This explains the weak, temperature-independent paramagnetism observed in alkali metals.
This behavior is a direct consequence of the Pauli exclusion principle, as only electrons near $ε_{F}$ can respond to the magnetic field.

We have now concluded our study of the ideal Fermi gas, having explored its equation of state, thermal properties, and magnetic response. In the next module, we will turn our attention to the fascinating world of ideal Bose gases, where the quantum statistics lead to a dramatic collective phenomenon: Bose-Einstein condensation.

Recommended resources

1. Electron Spin and Magnetic Fields

2. Magnetization at Zero Temperature

3. Rigorous Calculation

4. Finite Temperature Correction

Conclusion